Theorem ertr2d 8708

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

Hypotheses

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

Assertion

Ref	Expression
ertr2d	⊢ (𝜑 → 𝐶𝑅𝐴)

Proof of Theorem ertr2d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ertrd.5	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		ertrd.6	. . 3 ⊢ (𝜑 → 𝐵𝑅𝐶)
4	1, 2, 3	ertrd 8707	. 2 ⊢ (𝜑 → 𝐴𝑅𝐶)
5	1, 4	ersym 8703	1 ⊢ (𝜑 → 𝐶𝑅𝐴)

Syntax hints:   → wi 4   class class class wbr 5109   Er wer 8687

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 5257  ax-pr 5404

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 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-er 8690

This theorem is referenced by:  pi1xfrcnvlem  25215