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Theorem tfisi 4508
Description: A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Hypotheses
Ref Expression
tfisi.a (𝜑𝐴𝑉)
tfisi.b (𝜑𝑇 ∈ On)
tfisi.c ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
tfisi.d (𝑥 = 𝑦 → (𝜓𝜒))
tfisi.e (𝑥 = 𝐴 → (𝜓𝜃))
tfisi.f (𝑥 = 𝑦𝑅 = 𝑆)
tfisi.g (𝑥 = 𝐴𝑅 = 𝑇)
Assertion
Ref Expression
tfisi (𝜑𝜃)
Distinct variable groups:   𝑥,𝑦,𝑇   𝑦,𝑅   𝑥,𝑆   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦   𝑥,𝐴   𝜃,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem tfisi
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3121 . 2 𝑇𝑇
2 eqid 2140 . . . . 5 𝑇 = 𝑇
3 tfisi.a . . . . . 6 (𝜑𝐴𝑉)
4 tfisi.b . . . . . . 7 (𝜑𝑇 ∈ On)
5 eqeq2 2150 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑅 = 𝑧𝑅 = 𝑤))
6 sseq1 3124 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (𝑧𝑇𝑤𝑇))
76anbi2d 460 . . . . . . . . . . . 12 (𝑧 = 𝑤 → ((𝜑𝑧𝑇) ↔ (𝜑𝑤𝑇)))
87imbi1d 230 . . . . . . . . . . 11 (𝑧 = 𝑤 → (((𝜑𝑧𝑇) → 𝜓) ↔ ((𝜑𝑤𝑇) → 𝜓)))
95, 8imbi12d 233 . . . . . . . . . 10 (𝑧 = 𝑤 → ((𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ (𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓))))
109albidv 1797 . . . . . . . . 9 (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓))))
11 tfisi.f . . . . . . . . . . . 12 (𝑥 = 𝑦𝑅 = 𝑆)
1211eqeq1d 2149 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑅 = 𝑤𝑆 = 𝑤))
13 tfisi.d . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝜓𝜒))
1413imbi2d 229 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝜑𝑤𝑇) → 𝜓) ↔ ((𝜑𝑤𝑇) → 𝜒)))
1512, 14imbi12d 233 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓)) ↔ (𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))))
1615cbvalv 1890 . . . . . . . . 9 (∀𝑥(𝑅 = 𝑤 → ((𝜑𝑤𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)))
1710, 16syl6bb 195 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))))
18 eqeq2 2150 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑅 = 𝑧𝑅 = 𝑇))
19 sseq1 3124 . . . . . . . . . . . 12 (𝑧 = 𝑇 → (𝑧𝑇𝑇𝑇))
2019anbi2d 460 . . . . . . . . . . 11 (𝑧 = 𝑇 → ((𝜑𝑧𝑇) ↔ (𝜑𝑇𝑇)))
2120imbi1d 230 . . . . . . . . . 10 (𝑧 = 𝑇 → (((𝜑𝑧𝑇) → 𝜓) ↔ ((𝜑𝑇𝑇) → 𝜓)))
2218, 21imbi12d 233 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ (𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓))))
2322albidv 1797 . . . . . . . 8 (𝑧 = 𝑇 → (∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓))))
24 simp3l 1010 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝜑)
25 simp2 983 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅 = 𝑧)
26 simp1l 1006 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑧 ∈ On)
2725, 26eqeltrd 2217 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅 ∈ On)
28 simp3r 1011 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑧𝑇)
2925, 28eqsstrd 3137 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝑅𝑇)
30 simpl3l 1037 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝜑)
31 simpl1l 1033 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑧 ∈ On)
32 simpr 109 . . . . . . . . . . . . . . . . . . 19 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑅)
33 simpl2 986 . . . . . . . . . . . . . . . . . . 19 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑅 = 𝑧)
3432, 33eleqtrd 2219 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑧)
35 onelss 4316 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ On → (𝑣 / 𝑥𝑅𝑧𝑣 / 𝑥𝑅𝑧))
3631, 34, 35sylc 62 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑧)
37 simpl3r 1038 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑧𝑇)
3836, 37sstrd 3111 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅𝑇)
39 eqeq2 2150 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 / 𝑥𝑅 → (𝑆 = 𝑤𝑆 = 𝑣 / 𝑥𝑅))
40 sseq1 3124 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑣 / 𝑥𝑅 → (𝑤𝑇𝑣 / 𝑥𝑅𝑇))
4140anbi2d 460 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑣 / 𝑥𝑅 → ((𝜑𝑤𝑇) ↔ (𝜑𝑣 / 𝑥𝑅𝑇)))
4241imbi1d 230 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑣 / 𝑥𝑅 → (((𝜑𝑤𝑇) → 𝜒) ↔ ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)))
4339, 42imbi12d 233 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑣 / 𝑥𝑅 → ((𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) ↔ (𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒))))
4443albidv 1797 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑣 / 𝑥𝑅 → (∀𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) ↔ ∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒))))
45 simpl1r 1034 . . . . . . . . . . . . . . . . . 18 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)))
4644, 45, 34rspcdva 2797 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)))
47 eqidd 2141 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → 𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅)
48 nfcv 2282 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝑦
49 nfcv 2282 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝑆
5048, 49, 11csbhypf 3042 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑦𝑣 / 𝑥𝑅 = 𝑆)
5150eqcomd 2146 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑦𝑆 = 𝑣 / 𝑥𝑅)
5251equcoms 1685 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣𝑆 = 𝑣 / 𝑥𝑅)
5352eqeq1d 2149 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑆 = 𝑣 / 𝑥𝑅𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅))
54 nfv 1509 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥𝜒
5554, 13sbhypf 2738 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = 𝑦 → ([𝑣 / 𝑥]𝜓𝜒))
5655bicomd 140 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = 𝑦 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
5756equcoms 1685 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
5857imbi2d 229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒) ↔ ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓)))
5953, 58imbi12d 233 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 → ((𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)) ↔ (𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓))))
6059spv 1833 . . . . . . . . . . . . . . . . 17 (∀𝑦(𝑆 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → 𝜒)) → (𝑣 / 𝑥𝑅 = 𝑣 / 𝑥𝑅 → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓)))
6146, 47, 60sylc 62 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → ((𝜑𝑣 / 𝑥𝑅𝑇) → [𝑣 / 𝑥]𝜓))
6230, 38, 61mp2and 430 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) ∧ 𝑣 / 𝑥𝑅𝑅) → [𝑣 / 𝑥]𝜓)
6362ex 114 . . . . . . . . . . . . . 14 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → (𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓))
6463alrimiv 1847 . . . . . . . . . . . . 13 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → ∀𝑣(𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓))
6550eleq1d 2209 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → (𝑣 / 𝑥𝑅𝑅𝑆𝑅))
6665, 55imbi12d 233 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ((𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓) ↔ (𝑆𝑅𝜒)))
6766cbvalv 1890 . . . . . . . . . . . . 13 (∀𝑣(𝑣 / 𝑥𝑅𝑅 → [𝑣 / 𝑥]𝜓) ↔ ∀𝑦(𝑆𝑅𝜒))
6864, 67sylib 121 . . . . . . . . . . . 12 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → ∀𝑦(𝑆𝑅𝜒))
69 tfisi.c . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
7024, 27, 29, 68, 69syl121anc 1222 . . . . . . . . . . 11 (((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑𝑧𝑇)) → 𝜓)
71703exp 1181 . . . . . . . . . 10 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) → (𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)))
7271alrimiv 1847 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒))) → ∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓)))
7372ex 114 . . . . . . . 8 (𝑧 ∈ On → (∀𝑤𝑧𝑦(𝑆 = 𝑤 → ((𝜑𝑤𝑇) → 𝜒)) → ∀𝑥(𝑅 = 𝑧 → ((𝜑𝑧𝑇) → 𝜓))))
7417, 23, 73tfis3 4507 . . . . . . 7 (𝑇 ∈ On → ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)))
754, 74syl 14 . . . . . 6 (𝜑 → ∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)))
76 tfisi.g . . . . . . . . 9 (𝑥 = 𝐴𝑅 = 𝑇)
7776eqeq1d 2149 . . . . . . . 8 (𝑥 = 𝐴 → (𝑅 = 𝑇𝑇 = 𝑇))
78 tfisi.e . . . . . . . . 9 (𝑥 = 𝐴 → (𝜓𝜃))
7978imbi2d 229 . . . . . . . 8 (𝑥 = 𝐴 → (((𝜑𝑇𝑇) → 𝜓) ↔ ((𝜑𝑇𝑇) → 𝜃)))
8077, 79imbi12d 233 . . . . . . 7 (𝑥 = 𝐴 → ((𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)) ↔ (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃))))
8180spcgv 2776 . . . . . 6 (𝐴𝑉 → (∀𝑥(𝑅 = 𝑇 → ((𝜑𝑇𝑇) → 𝜓)) → (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃))))
823, 75, 81sylc 62 . . . . 5 (𝜑 → (𝑇 = 𝑇 → ((𝜑𝑇𝑇) → 𝜃)))
832, 82mpi 15 . . . 4 (𝜑 → ((𝜑𝑇𝑇) → 𝜃))
8483expd 256 . . 3 (𝜑 → (𝜑 → (𝑇𝑇𝜃)))
8584pm2.43i 49 . 2 (𝜑 → (𝑇𝑇𝜃))
861, 85mpi 15 1 (𝜑𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963  wal 1330   = wceq 1332  wcel 1481  [wsb 1736  wral 2417  csb 3006  wss 3075  Oncon0 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-in 3081  df-ss 3088  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297
This theorem is referenced by: (None)
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