Theorem copsex2b 37570

Description: Biconditional form of copsex2d 37569. TODO: prove a relative version, that is, with ∃𝑥 ∈ 𝑉∃𝑦 ∈ 𝑊...(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊). (Contributed by BJ, 27-Dec-2023.)

Hypotheses

Ref	Expression
copsex2b.xph	⊢ (𝜑 → ∀𝑥𝜑)
copsex2b.yph	⊢ (𝜑 → ∀𝑦𝜑)
copsex2b.xch	⊢ (𝜑 → Ⅎ𝑥𝜒)
copsex2b.ych	⊢ (𝜑 → Ⅎ𝑦𝜒)
copsex2b.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
copsex2b	⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem copsex2b

Step	Hyp	Ref	Expression
1		eqcom 2759	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3448	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	opth 5434	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
5	1, 4	bitri 277	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		eqvisset 3464	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
7		eqvisset 3464	. . . . . . 7 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
8	6, 7	anim12i 621	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	5, 8	sylbi 219	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	9	adantr 483	. . . 4 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	10	exlimivv 1942	. . 3 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
12	11	anim2i 625	. 2 ⊢ ((𝜑 ∧ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → (𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
13		simpl 485	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	13	anim2i 625	. 2 ⊢ ((𝜑 ∧ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) → (𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
15		copsex2b.xph	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
16		ax-5 1920	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V))
17	15, 16	hban 2324	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ∀𝑥(𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
18		copsex2b.yph	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
19		ax-5 1920	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑦(𝐴 ∈ V ∧ 𝐵 ∈ V))
20	18, 19	hban 2324	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ∀𝑦(𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
21		copsex2b.xch	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
22	21	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → Ⅎ𝑥𝜒)
23		copsex2b.ych	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜒)
24	23	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → Ⅎ𝑦𝜒)
25		simprl 778	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐴 ∈ V)
26		simprr 780	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐵 ∈ V)
27		copsex2b.is	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
28	27	adantlr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
29	17, 20, 22, 24, 25, 26, 28	copsex2d 37569	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
30		ibar 535	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜒 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
31	30	adantl 484	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝜒 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
32	29, 31	bitrd 281	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
33	12, 14, 32	pm5.21nd 809	1 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1548   = wceq 1550  ∃wex 1789  Ⅎwnf 1793   ∈ wcel 2132  Vcvv 3444  ⟨cop 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579

This theorem is referenced by:  opelopabb  37572