Theorem eleq12 2820

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 2818	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2		eleq2 2819	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
3	1, 2	sylan9bb 513	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2728  df-clel 2809

This theorem is referenced by:  rru  3681  trel  5153  epelg  5446  preleqg  9208  preleqALT  9210  oemapval  9276  cantnf  9286  wemapwe  9290  nnsdomel  9571  cldval  21874  isufil  22754  umgr2v2enb1  27568  issiga  31746  bj-epelg  34924  rdgssun  35235  fvineqsneu  35268  matunitlindf  35461  wepwsolem  40511  aomclem8  40530  grumnud  41518  nelbr  44381