Theorem ertr3d 6651

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

Hypotheses

Ref	Expression
ersymb.1	|- ( ph -> R Er X )
ertr3d.5	|- ( ph -> B R A )
ertr3d.6	|- ( ph -> B R C )

Assertion

Ref	Expression
ertr3d	|- ( ph -> A R C )

Proof of Theorem ertr3d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 |- ( ph -> R Er X )
2		ertr3d.5	. . 3 |- ( ph -> B R A )
3	1, 2	ersym 6645	. 2 |- ( ph -> A R B )
4		ertr3d.6	. 2 |- ( ph -> B R C )
5	1, 3, 4	ertrd 6649	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   class class class wbr 4051   Er wer 6630

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-er 6633

This theorem is referenced by:  xmetresbl  14987