Theorem eleq2s 1983

Description: Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
eleq2s.1	|- (A e. B -> ph)
eleq2s.2	|- C = B

Assertion

Ref	Expression
eleq2s	|- (A e. C -> ph)

Proof of Theorem eleq2s

Step	Hyp	Ref	Expression
1		eleq2s.2	. . 3 |- C = B
2	1	eleq2i 1961	. 2 |- (A e. C <-> A e. B)
3		eleq2s.1	. 2 |- (A e. B -> ph)
4	2, 3	sylbi 216	1 |- (A e. C -> ph)

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

This theorem is referenced by:  bnj1173 13441  bnj1255 13466  bnj1258 13468  bnj1296 13484  bnj1498 13562  repcpwti 14503

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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