Theorem eleq12 2828

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 2826	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2		eleq2 2827	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
3	1, 2	sylan9bb 509	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-clel 2813

This theorem is referenced by:  rru  3787  trel  5273  epelg  5589  preleqg  9652  preleqALT  9654  oemapval  9720  cantnf  9730  wemapwe  9734  nnsdomel  10027  cldval  23046  isufil  23926  taylthlem2  26430  umgr2v2enb1  29558  issiga  34092  bj-epelg  37050  rdgssun  37360  matunitlindf  37604  wepwsolem  43030  aomclem8  43049  grumnud  44281  nelbr  47223