Theorem ertrd 6796

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

Hypotheses

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

Assertion

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

Proof of Theorem ertrd

Step	Hyp	Ref	Expression
1		ertrd.5	. 2 |- ( ph -> A R B )
2		ertrd.6	. 2 |- ( ph -> B R C )
3		ersymb.1	. . 3 |- ( ph -> R Er X )
4	3	ertr 6795	. 2 |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) )
5	1, 2, 4	mp2and 433	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 4114   Er wer 6777

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 4233  ax-pow 4292  ax-pr 4327

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 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-co 4763  df-er 6780

This theorem is referenced by:  ertr2d  6797  ertr3d  6798  ertr4d  6799  erinxp  6856