Theorem syl5breq 2647

Description: A chained equality inference for a binary relation.

Hypotheses

Ref	Expression
syl5breq.1	|- (ph -> A = B)
syl5breq.2	|- CRA

Assertion

Ref	Expression
syl5breq	|- (ph -> CRB)

Proof of Theorem syl5breq

Step	Hyp	Ref	Expression
1		syl5breq.2	. . 3 |- CRA
2	1	a1i 8	. 2 |- (ph -> CRA)
3		syl5breq.1	. 2 |- (ph -> A = B)
4	2, 3	breqtrd 2636	1 |- (ph -> CRB)

Syntax hints:   -> wi 3   = wceq 955   class class class wbr 2616

This theorem is referenced by:  syl5breqr 2648  phplem3 4503  sqrge0 6653  blocnilem 8448  siilem1 8495  nmcoplb 9949  nmbdfnlb 9969  nmcfnlb 9978  unierr 10028  leoprf2t 10051  leoprft 10052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-un 2048  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617

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