Theorem ertr4d 6788

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
ersymb.1	|- ( ph -> R Er X )
ertr4d.5	|- ( ph -> A R B )
ertr4d.6	|- ( ph -> C R B )

Assertion

Ref	Expression
ertr4d	|- ( ph -> A R C )

Proof of Theorem ertr4d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 |- ( ph -> R Er X )
2		ertr4d.5	. 2 |- ( ph -> A R B )
3		ertr4d.6	. . 3 |- ( ph -> C R B )
4	1, 3	ersym 6781	. 2 |- ( ph -> B R C )
5	1, 2, 4	ertrd 6785	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 4111   Er wer 6766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-er 6769

This theorem is referenced by:  erref  6789