Theorem ertrd 8740

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
ertrd	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem ertrd

Step	Hyp	Ref	Expression
1		ertrd.5	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		ertrd.6	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
3		ersymb.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
4	3	ertr 8739	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 699	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5124   Er wer 8721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-co 5668  df-er 8724

This theorem is referenced by:  ertr2d  8741  ertr3d  8742  ertr4d  8743  erinxp  8810  nqereq  10954  adderpq  10975  mulerpq  10976  efgred2  19739  efgcpbllemb  19741  efgcpbl2  19743  pcophtb  24985  pi1xfr  25011  pi1xfrcnvlem  25012  erbr3b  32602  prjspner1  42616