Theorem opeqex 4067

Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Assertion

Ref	Expression
opeqex	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Proof of Theorem opeqex

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2151	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ ⟨𝐶, 𝐷⟩))
2	1	exbidv 1753	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ ∃𝑥 𝑥 ∈ ⟨𝐶, 𝐷⟩))
3		opm 4052	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		opm 4052	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐶, 𝐷⟩ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	2, 3, 4	3bitr3g 220	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1289  ∃wex 1426   ∈ wcel 1438  Vcvv 2619  ⟨cop 3444

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450

This theorem is referenced by:  epelg  4108