Theorem ertr4d 8715

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

Hypotheses

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ertr4d.5	⊢ (𝜑 → 𝐴𝑅𝐵)
ertr4d.6	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
ertr4d	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem ertr4d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ertr4d.5	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		ertr4d.6	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
4	1, 3	ersym 8708	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 2, 4	ertrd 8712	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5110   Er wer 8692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-er 8695

This theorem is referenced by:  erref  8716  erdisj  8753  nqereu  10915  nqereq  10921  efgredeu  19823  pi1xfr  25195  pi1xfrcnvlem  25196  prjspner1  43338