Theorem syl6eqel 2441

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl6eqel.1	⊢ (φ → A = B)
syl6eqel.2	⊢ B ∈ C

Assertion

Ref	Expression
syl6eqel	⊢ (φ → A ∈ C)

Proof of Theorem syl6eqel

Step	Hyp	Ref	Expression
1		syl6eqel.1	. 2 ⊢ (φ → A = B)
2		syl6eqel.2	. . 3 ⊢ B ∈ C
3	2	a1i 10	. 2 ⊢ (φ → B ∈ C)
4	1, 3	eqeltrd 2427	1 ⊢ (φ → A ∈ C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  syl6eqelr  2442  snex  4112  sikss1c1c  4268  ins2kss  4280  ins3kss  4281  iotaex  4357  eladdc  4399  ssfin  4471  f0cli  5419  elimdelov  5574  ndmovcl  5615  brfns  5834  muccl  6133  ncaddccl  6145  ceclb  6184  cet  6235  nclenn  6250  nchoicelem12  6301  nchoicelem17  6306  nchoicelem18  6307  nchoicelem19  6308