Theorem ertrd 8662

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 8661	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 700	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5100   Er wer 8642

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-co 5641  df-er 8645

This theorem is referenced by:  ertr2d  8663  ertr3d  8664  ertr4d  8665  erinxp  8740  nqereq  10858  adderpq  10879  mulerpq  10880  efgred2  19694  efgcpbllemb  19696  efgcpbl2  19698  pcophtb  24997  pi1xfr  25023  pi1xfrcnvlem  25024  erbr3b  32707  prjspner1  42984  chnerlem1  47240