Theorem ertr4d 8663

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 8656	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 2, 4	ertrd 8660	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5085   Er wer 8640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-er 8643

This theorem is referenced by:  erref  8664  erdisj  8701  nqereu  10852  nqereq  10858  efgredeu  19727  pi1xfr  25022  pi1xfrcnvlem  25023  prjspner1  43059