Theorem syl6ssr 2111

Description: A chained subclass and equality deduction.

Hypotheses

Ref	Expression
syl6ssr.1	|- (ph -> A (_ B)
syl6ssr.2	|- C = B

Assertion

Ref	Expression
syl6ssr	|- (ph -> A (_ C)

Proof of Theorem syl6ssr

Step	Hyp	Ref	Expression
1		syl6ssr.1	. 2 |- (ph -> A (_ B)
2		syl6ssr.2	. . 3 |- C = B
3	2	eqcomi 1482	. 2 |- B = C
4	1, 3	syl6ss 2110	1 |- (ph -> A (_ C)

Syntax hints:   -> wi 3   = wceq 958   (_ wss 2050

This theorem is referenced by:  iunpw 2920  tfrlem9 3925  tfrlem13 3929  tz7.49 3965  cplem1 4730  zorn2lem2 4799  infxpidmlem5 7557  eltopss 7604  elcls3 7708  mdsymlem1 10325  idhme 10508

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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