Theorem copsex2gb 5679

Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5680. (Contributed by NM, 12-Mar-2014.)

Hypothesis

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

Assertion

Ref	Expression
copsex2gb	⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))

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

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem copsex2gb

Step	Hyp	Ref	Expression
1		elvv 5626	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 625	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		19.41vv 1951	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		copsex2ga.1	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
5	4	pm5.32i 577	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
6	5	2exbii 1849	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
7	2, 3, 6	3bitr2ri 302	1 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   × cxp 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561

This theorem is referenced by:  copsex2ga  5680  elopaba  5681  dfxrn2  35643  elcnvlem  39981