Theorem ertr2d 8728

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 8727	. 2 ⊢ (𝜑 → 𝐴𝑅𝐶)
5	1, 4	ersym 8723	1 ⊢ (𝜑 → 𝐶𝑅𝐴)

This proof depends on syntax axioms:   → wi 4   class class class wbr 5103   Er wer 8707

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-er 8710

This theorem is used by:  pi1xfrcnvlem  25370