Theorem syl6eqelr 2442

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

Hypotheses

Ref	Expression
syl6eqelr.1	⊢ (φ → B = A)
syl6eqelr.2	⊢ B ∈ C

Assertion

Ref	Expression
syl6eqelr	⊢ (φ → A ∈ C)

Proof of Theorem syl6eqelr

Step	Hyp	Ref	Expression
1		syl6eqelr.1	. . 3 ⊢ (φ → B = A)
2	1	eqcomd 2358	. 2 ⊢ (φ → A = B)
3		syl6eqelr.2	. 2 ⊢ B ∈ C
4	2, 3	syl6eqel 2441	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:  cnvkexg  4287  p6exg  4291  ssetkex  4295  sikexg  4297  ins2kexg  4306  ins3kexg  4307  0cnelphi  4598  mapprc  6005  enmap1lem5  6074  nenpw1pwlem2  6086  sbthlem3  6206