Theorem ertr2d 8156

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 8155	. 2 ⊢ (𝜑 → 𝐴𝑅𝐶)
5	1, 4	ersym 8151	1 ⊢ (𝜑 → 𝐶𝑅𝐴)

Syntax hints:   → wi 4   class class class wbr 4962   Er wer 8136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-er 8139

This theorem is referenced by:  pi1xfrcnvlem  23343