Theorem copsex2gb 5230

Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5231. (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 5177	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 731	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		19.41vv 1915	. 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 669	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
6	5	2exbii 1775	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
7	2, 3, 6	3bitr2ri 289	1 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  copsex2ga  5231  elopaba  5232  elcnvlem  37907