Theorem eleq12 2099

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 2097	. 2 ⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶))
2		eleq2 2098	. 2 ⊢ (𝐶 = 𝐷 → (B ∈ 𝐶 ↔ B ∈ 𝐷))
3	1, 2	sylan9bb 435	1 ⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  trel  3852  pwnss  3903  epelg  4018  preleq  4233  acexmid  5454