Theorem eleq12 2874

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 2872	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2		eleq2 2873	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
3	1, 2	sylan9bb 510	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-cleq 2790  df-clel 2865

This theorem is referenced by:  rru  3709  trel  5077  epelg  5361  epelgOLD  5362  preleqg  8931  preleqALT  8933  oemapval  8999  cantnf  9009  wemapwe  9013  nnsdomel  9272  cldval  21319  isufil  22199  umgr2v2enb1  26995  issiga  30984  bj-elep  33979  rdgssun  34211  fvineqsneu  34244  matunitlindf  34442  wepwsolem  39148  aomclem8  39167  grumnud  40140  nelbr  43011