Theorem ertr4d 8654

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 8647	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 2, 4	ertrd 8651	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5098   Er wer 8632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-er 8635

This theorem is referenced by:  erref  8655  erdisj  8692  nqereu  10840  nqereq  10846  efgredeu  19681  pi1xfr  25011  pi1xfrcnvlem  25012  prjspner1  42865