Theorem syl5eleq 2439

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

Hypotheses

Ref	Expression
syl5eleq.1	⊢ A ∈ B
syl5eleq.2	⊢ (φ → B = C)

Assertion

Ref	Expression
syl5eleq	⊢ (φ → A ∈ C)

Proof of Theorem syl5eleq

Step	Hyp	Ref	Expression
1		syl5eleq.1	. . 3 ⊢ A ∈ B
2	1	a1i 10	. 2 ⊢ (φ → A ∈ B)
3		syl5eleq.2	. 2 ⊢ (φ → B = C)
4	2, 3	eleqtrd 2429	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:  syl5eleqr  2440  enadj  6061