Theorem syl5ssr 1545

Description: A chained subclass and equality deduction.

Hypotheses

Ref	Expression
syl5ssr.1	|- (ph -> A (_ B)
syl5ssr.2	|- A = C

Assertion

Ref	Expression
syl5ssr	|- (ph -> C (_ B)

Proof of Theorem syl5ssr

Step	Hyp	Ref	Expression
1		syl5ssr.1	. 2 |- (ph -> A (_ B)
2		syl5ssr.2	. . 3 |- A = C
3	2	cleqcomi 1105	. 2 |- C = A
4	1, 3	syl5ss 1544	1 |- (ph -> C (_ B)

Syntax hints:   -> wi 2   = wceq 1091   (_ wss 1487

This theorem is referenced by:  dmen 3310  cardprc 3667  alephgeom 3687

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

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