Theorem eleq12 2859

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 2857	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2		eleq2 2858	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
3	1, 2	sylan9bb 518	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-clel 2844

This theorem is referenced by:  rru  3751  trel  5230  epelg  5563  preleqg  9583  preleqALT  9585  oemapval  9651  cantnf  9661  wemapwe  9665  nnsdomel  9975  cldval  23148  isufil  24028  taylthlem2  26502  umgr2v2enb1  29816  issiga  34446  bj-epelg  37591  rdgssun  37911  matunitlindf  38156  wepwsolem  43660  aomclem8  43679  grumnud  44887  nelbr  47899