Theorem copsex2ga 5805

Description: Implicit substitution inference for ordered pairs. Compare copsex2g 5492. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
copsex2ga.1	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
copsex2ga	⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))

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

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2ga

Step	Hyp	Ref	Expression
1		xpss 5691	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3977	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		copsex2ga.1	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
4	3	copsex2gb 5804	. . 3 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
5	4	baibr 537	. 2 ⊢ (𝐴 ∈ (V × V) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
6	2, 5	syl 17	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474  ⟨cop 4633   × cxp 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681

This theorem is referenced by: (None)