Theorem opeqex 4227

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 2230	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ ⟨𝐶, 𝐷⟩))
2	1	exbidv 1813	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ ∃𝑥 𝑥 ∈ ⟨𝐶, 𝐷⟩))
3		opm 4212	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		opm 4212	. 2 ⊢ (∃𝑥 𝑥 ∈ ⟨𝐶, 𝐷⟩ ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	2, 3, 4	3bitr3g 221	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  epelg  4268