Theorem ertrd 8716

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 8715	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 696	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5139   Er wer 8697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-co 5676  df-er 8700

This theorem is referenced by:  ertr2d  8717  ertr3d  8718  ertr4d  8719  erinxp  8782  nqereq  10927  adderpq  10948  mulerpq  10949  efgred2  19669  efgcpbllemb  19671  efgcpbl2  19673  pcophtb  24900  pi1xfr  24926  pi1xfrcnvlem  24927  erbr3b  32340  prjspner1  41918