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  4111  sikss1c1c  4267  ins2kss  4279  ins3kss  4280  iotaex  4356  eladdc  4398  ssfin  4470  f0cli  5418  elimdelov  5573  ndmovcl  5614  brfns  5833  muccl  6132  ncaddccl  6144  ceclb  6183  cet  6234  nclenn  6249  nchoicelem12  6300  nchoicelem17  6305  nchoicelem18  6306  nchoicelem19  6307