Theorem copsex2gb 5791

Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5792. (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 5734	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2	1	anbi1i 636	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3		19.41vv 1983	. 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 585	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
6	5	2exbii 1882	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
7	2, 3, 6	3bitr2ri 303	1 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   × cxp 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-un 3907  df-in 3909  df-ss 3919  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-xp 5665

This theorem is used by:  copsex2ga  5792  elopaba  5793  dfxrn2  39135  elcnvlem  44443