Theorem eleq12 2084

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Assertion

Ref	Expression
eleq12	⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷))

Proof of Theorem eleq12

Step	Hyp	Ref	Expression
1		eleq1 2082	. 2 ⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶))
2		eleq2 2083	. 2 ⊢ (𝐶 = 𝐷 → (B ∈ 𝐶 ↔ B ∈ 𝐷))
3	1, 2	sylan9bb 438	1 ⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018

This theorem is referenced by:  trel  3835  pwnss  3886  epelg  4001  preleq  4217  acexmid  5435