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Theorem br4 31979
Description: Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
Hypotheses
Ref Expression
br4.1 (𝑎 = 𝐴 → (𝜑𝜓))
br4.2 (𝑏 = 𝐵 → (𝜓𝜒))
br4.3 (𝑐 = 𝐶 → (𝜒𝜃))
br4.4 (𝑑 = 𝐷 → (𝜃𝜏))
br4.5 (𝑥 = 𝑋𝑃 = 𝑄)
br4.6 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}
Assertion
Ref Expression
br4 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥,𝐴   𝐵,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜒,𝑏   𝑄,𝑎,𝑏,𝑐,𝑑,𝑥   𝐶,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝐷,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜓,𝑎   𝑋,𝑎,𝑏,𝑐,𝑑,𝑥   𝑃,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞   𝑆,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜏,𝑎,𝑏,𝑐,𝑑,𝑥   𝜃,𝑐   𝜑,𝑝,𝑞,𝑥
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑞,𝑝,𝑏,𝑐,𝑑)   𝜒(𝑥,𝑞,𝑝,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑞,𝑝,𝑎,𝑏,𝑑)   𝜏(𝑞,𝑝)   𝑃(𝑥)   𝑄(𝑞,𝑝)   𝑅(𝑥,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝑋(𝑞,𝑝)

Proof of Theorem br4
StepHypRef Expression
1 opex 5133 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5133 . . 3 𝐶, 𝐷⟩ ∈ V
3 eqeq1 2821 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
433anbi1d 1557 . . . . . 6 (𝑝 = ⟨𝐴, 𝐵⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
54rexbidv 3251 . . . . 5 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
652rexbidv 3256 . . . 4 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
762rexbidv 3256 . . 3 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
8 eqeq1 2821 . . . . . . 7 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
983anbi2d 1558 . . . . . 6 (𝑞 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
109rexbidv 3251 . . . . 5 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
11102rexbidv 3256 . . . 4 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
12112rexbidv 3256 . . 3 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
13 br4.6 . . 3 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}
141, 2, 7, 12, 13brab 5204 . 2 (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
15 vex 3405 . . . . . . . . . . . 12 𝑎 ∈ V
16 vex 3405 . . . . . . . . . . . 12 𝑏 ∈ V
1715, 16opth 5145 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵))
18 br4.1 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝜑𝜓))
19 br4.2 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝜓𝜒))
2018, 19sylan9bb 501 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜒))
2117, 20sylbi 208 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → (𝜑𝜒))
2221eqcoms 2825 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
23 vex 3405 . . . . . . . . . . . 12 𝑐 ∈ V
24 vex 3405 . . . . . . . . . . . 12 𝑑 ∈ V
2523, 24opth 5145 . . . . . . . . . . 11 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶𝑑 = 𝐷))
26 br4.3 . . . . . . . . . . . 12 (𝑐 = 𝐶 → (𝜒𝜃))
27 br4.4 . . . . . . . . . . . 12 (𝑑 = 𝐷 → (𝜃𝜏))
2826, 27sylan9bb 501 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝜒𝜏))
2925, 28sylbi 208 . . . . . . . . . 10 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ → (𝜒𝜏))
3029eqcoms 2825 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ → (𝜒𝜏))
3122, 30sylan9bb 501 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩) → (𝜑𝜏))
3231biimp3a 1586 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)
3332a1i 11 . . . . . 6 (((((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) ∧ (𝑏𝑃𝑐𝑃)) ∧ 𝑑𝑃) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3433rexlimdva 3230 . . . . 5 ((((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) ∧ (𝑏𝑃𝑐𝑃)) → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3534rexlimdvva 3237 . . . 4 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3635rexlimdvva 3237 . . 3 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
37 simpl1 1235 . . . . 5 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝑋𝑆)
38 simpl2l 1290 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐴𝑄)
39 simpl2r 1292 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐵𝑄)
40 simpl3l 1294 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐶𝑄)
41 simpl3r 1296 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐷𝑄)
42 eqidd 2818 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
43 eqidd 2818 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
44 simpr 473 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝜏)
45 opeq1 4606 . . . . . . . . . 10 (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
4645eqeq2d 2827 . . . . . . . . 9 (𝑐 = 𝐶 → (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩))
4746, 263anbi23d 1556 . . . . . . . 8 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃)))
48 opeq2 4607 . . . . . . . . . 10 (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
4948eqeq2d 2827 . . . . . . . . 9 (𝑑 = 𝐷 → (⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
5049, 273anbi23d 1556 . . . . . . . 8 (𝑑 = 𝐷 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏)))
5147, 50rspc2ev 3528 . . . . . . 7 ((𝐶𝑄𝐷𝑄 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏)) → ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))
5240, 41, 42, 43, 44, 51syl113anc 1494 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))
53 opeq1 4606 . . . . . . . . . 10 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5453eqeq2d 2827 . . . . . . . . 9 (𝑎 = 𝐴 → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩))
5554, 183anbi13d 1555 . . . . . . . 8 (𝑎 = 𝐴 → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓)))
56552rexbidv 3256 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓)))
57 opeq2 4607 . . . . . . . . . 10 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
5857eqeq2d 2827 . . . . . . . . 9 (𝑏 = 𝐵 → (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
5958, 193anbi13d 1555 . . . . . . . 8 (𝑏 = 𝐵 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)))
60592rexbidv 3256 . . . . . . 7 (𝑏 = 𝐵 → (∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)))
6156, 60rspc2ev 3528 . . . . . 6 ((𝐴𝑄𝐵𝑄 ∧ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)) → ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
6238, 39, 52, 61syl3anc 1483 . . . . 5 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
63 br4.5 . . . . . . 7 (𝑥 = 𝑋𝑃 = 𝑄)
6463rexeqdv 3345 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6563, 64rexeqbidv 3353 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6663, 65rexeqbidv 3353 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6763, 66rexeqbidv 3353 . . . . . 6 (𝑥 = 𝑋 → (∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6867rspcev 3513 . . . . 5 ((𝑋𝑆 ∧ ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
6937, 62, 68syl2anc 575 . . . 4 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
7069ex 399 . . 3 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (𝜏 → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
7136, 70impbid 203 . 2 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ 𝜏))
7214, 71syl5bb 274 1 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wrex 3108  cop 4387   class class class wbr 4855  {copab 4917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pr 5107
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856  df-opab 4918
This theorem is referenced by: (None)
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