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Theorem bnj964 31559
Description: Technical lemma for bnj69 31624. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj964.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj964.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj964.5 (𝜓′[𝑝 / 𝑛]𝜓)
bnj964.8 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj964.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj964.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj964.96 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
bnj964.165 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Assertion
Ref Expression
bnj964 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛   𝐷,𝑖   𝑖,𝐺   𝑅,𝑓,𝑖,𝑛   𝑖,𝑋   𝑓,𝑝,𝑖   𝑦,𝑓,𝑖,𝑛   𝑖,𝑚   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑚,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑚,𝑝)   𝐺(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj964
StepHypRef Expression
1 nfv 2015 . . . 4 𝑖(𝑅 FrSe 𝐴𝑋𝐴)
2 bnj964.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32bnj1095 31398 . . . . . . 7 (𝜓 → ∀𝑖𝜓)
4 bnj964.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
53, 4bnj1096 31399 . . . . . 6 (𝜒 → ∀𝑖𝜒)
65nf5i 2199 . . . . 5 𝑖𝜒
7 nfv 2015 . . . . 5 𝑖 𝑛 = suc 𝑚
8 nfv 2015 . . . . 5 𝑖 𝑝 = suc 𝑛
96, 7, 8nf3an 2006 . . . 4 𝑖(𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)
101, 9nfan 2004 . . 3 𝑖((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 31320 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
12 bnj645 31366 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → suc 𝑖𝑝)
13 simp3 1174 . . . . . . . 8 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
1413bnj706 31370 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → 𝑝 = suc 𝑛)
15 eleq2 2895 . . . . . . . . 9 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
1615biimpac 472 . . . . . . . 8 ((suc 𝑖𝑝𝑝 = suc 𝑛) → suc 𝑖 ∈ suc 𝑛)
17 elsuci 6029 . . . . . . . . 9 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛))
18 eqcom 2832 . . . . . . . . . 10 (suc 𝑖 = 𝑛𝑛 = suc 𝑖)
1918orbi2i 943 . . . . . . . . 9 ((suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛) ↔ (suc 𝑖𝑛𝑛 = suc 𝑖))
2017, 19sylib 210 . . . . . . . 8 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛𝑛 = suc 𝑖))
2116, 20syl 17 . . . . . . 7 ((suc 𝑖𝑝𝑝 = suc 𝑛) → (suc 𝑖𝑛𝑛 = suc 𝑖))
2212, 14, 21syl2anc 581 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (suc 𝑖𝑛𝑛 = suc 𝑖))
23 df-3an 1115 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
24233anbi3i 1204 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛)))
25 bnj255 31320 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛)))
2624, 25bitr4i 270 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
27 bnj345 31329 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛) ↔ (suc 𝑖𝑛 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
28 bnj252 31318 . . . . . . . . . . 11 ((suc 𝑖𝑛 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
2926, 27, 283bitri 289 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
3011anbi2i 618 . . . . . . . . . 10 ((suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
3129, 30bitr4i 270 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)))
32 bnj964.96 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3331, 32sylbir 227 . . . . . . . 8 ((suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3433ex 403 . . . . . . 7 (suc 𝑖𝑛 → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
35 df-3an 1115 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
36353anbi3i 1204 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖)))
37 bnj255 31320 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖)))
3836, 37bitr4i 270 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
39 bnj345 31329 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖) ↔ (𝑛 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
40 bnj252 31318 . . . . . . . . . . 11 ((𝑛 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4138, 39, 403bitri 289 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4211anbi2i 618 . . . . . . . . . 10 ((𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4341, 42bitr4i 270 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)))
44 bnj964.165 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4543, 44sylbir 227 . . . . . . . 8 ((𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4645ex 403 . . . . . . 7 (𝑛 = suc 𝑖 → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4734, 46jaoi 890 . . . . . 6 ((suc 𝑖𝑛𝑛 = suc 𝑖) → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4822, 47mpcom 38 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4911, 48sylbir 227 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
50493expia 1156 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5110, 50alrimi 2258 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
52 bnj964.5 . . . . 5 (𝜓′[𝑝 / 𝑛]𝜓)
53 vex 3417 . . . . 5 𝑝 ∈ V
542, 52, 53bnj539 31507 . . . 4 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
55 bnj964.8 . . . 4 (𝜓″[𝐺 / 𝑓]𝜓′)
56 bnj964.12 . . . 4 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
57 bnj964.13 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
5854, 55, 56, 57bnj965 31558 . . 3 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5958bnj115 31340 . 2 (𝜓″ ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
6051, 59sylibr 226 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 880  w3a 1113  wal 1656   = wceq 1658  wcel 2166  wral 3117  [wsbc 3662  cun 3796  {csn 4397  cop 4403   ciun 4740  suc csuc 5965   Fn wfn 6118  cfv 6123  ωcom 7326  w-bnj17 31301   predc-bnj14 31303   FrSe w-bnj15 31307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-suc 5969  df-iota 6086  df-fv 6131  df-bnj17 31302
This theorem is referenced by:  bnj910  31564
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