Theorem syl6eleqr 2444

Description: A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Hypotheses

Ref	Expression
syl6eleqr.1	⊢ (φ → A ∈ B)
syl6eleqr.2	⊢ C = B

Assertion

Ref	Expression
syl6eleqr	⊢ (φ → A ∈ C)

Proof of Theorem syl6eleqr

Step	Hyp	Ref	Expression
1		syl6eleqr.1	. 2 ⊢ (φ → A ∈ B)
2		syl6eleqr.2	. . 3 ⊢ C = B
3	2	eqcomi 2357	. 2 ⊢ B = C
4	1, 3	syl6eleq 2443	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:  reiotacl2  4364  nnadjoinpw  4522  sfinltfin  4536  vinf  4556  nulnnn  4557  ecopqsi  5982  ncidg  6123