Theorem eleq12 2818

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 2816	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2		eleq2 2817	. 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 1540   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-clel 2803

This theorem is referenced by:  rru  3747  trel  5218  epelg  5532  preleqg  9544  preleqALT  9546  oemapval  9612  cantnf  9622  wemapwe  9626  nnsdomel  9919  cldval  22886  isufil  23766  taylthlem2  26258  umgr2v2enb1  29430  issiga  34075  bj-epelg  37029  rdgssun  37339  matunitlindf  37585  wepwsolem  43004  aomclem8  43023  grumnud  44248  nelbr  47248