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Theorem copsex2g 4008
Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
Hypothesis
Ref Expression
copsex2g.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
copsex2g ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2g
StepHypRef Expression
1 elisset 2583 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2583 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 eeanv 1821 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 nfe1 1399 . . . . 5 𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5 nfv 1435 . . . . 5 𝑥𝜓
64, 5nfbi 1495 . . . 4 𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
7 nfe1 1399 . . . . . . 7 𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
87nfex 1542 . . . . . 6 𝑦𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
9 nfv 1435 . . . . . 6 𝑦𝜓
108, 9nfbi 1495 . . . . 5 𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
11 opeq12 3576 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12 copsexg 4006 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1312eqcoms 2057 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1411, 13syl 14 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
15 copsex2g.1 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1614, 15bitr3d 183 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
1710, 16exlimi 1499 . . . 4 (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
186, 17exlimi 1499 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
193, 18sylbir 129 . 2 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
201, 2, 19syl2an 277 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  cop 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409
This theorem is referenced by:  opelopabga  4025  ov6g  5663  ltresr  6943
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