Theorem ertr3d 8739

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

Hypotheses

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ertr3d.5	⊢ (𝜑 → 𝐵𝑅𝐴)
ertr3d.6	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
ertr3d	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem ertr3d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ertr3d.5	. . 3 ⊢ (𝜑 → 𝐵𝑅𝐴)
3	1, 2	ersym 8733	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
4		ertr3d.6	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 3, 4	ertrd 8737	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

This proof depends on syntax axioms:   → wi 4   class class class wbr 5102   Er wer 8717

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-er 8720

This theorem is used by:  nqereq  11023  efgred2  19972  xmetresbl  24761  pcophtb  25355  pi1xfr  25381  pi1xfrcnvlem  25382  erbr3b  33258  prtlem10  40015