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Theorem copsex2gb 5386
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5387. (Contributed by NM, 12-Mar-2014.)
Hypothesis
Ref Expression
copsex2ga.1 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
copsex2gb (∃𝑥𝑦(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
StepHypRef Expression
1 elvv 5334 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
21anbi1i 733 . 2 ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 19.41vv 2027 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 copsex2ga.1 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
54pm5.32i 672 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
652exbii 1924 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
72, 3, 63bitr2ri 289 1 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cop 4327   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272
This theorem is referenced by:  copsex2ga  5387  elopaba  5388  dfxrn2  34461  elcnvlem  38409
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