Theorem syl5eleq 1977

Description: A membership and equality inference.

Hypotheses

Ref	Expression
syl5eleq.1	|- (ph -> A = B)
syl5eleq.2	|- C e. A

Assertion

Ref	Expression
syl5eleq	|- (ph -> C e. B)

Proof of Theorem syl5eleq

Step	Hyp	Ref	Expression
1		syl5eleq.2	. . 3 |- C e. A
2	1	a1i 8	. 2 |- (ph -> C e. A)
3		syl5eleq.1	. 2 |- (ph -> A = B)
4	2, 3	eleqtrd 1973	1 |- (ph -> C e. B)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300

This theorem is referenced by:  syl5eleqr 1978  eqelsuc 3750  tfrlem11 5129  oalimcl 5242  omlimcl 5257

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880

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