Theorem ertrd 8649

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 8648	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 699	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5096   Er wer 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-co 5631  df-er 8633

This theorem is referenced by:  ertr2d  8650  ertr3d  8651  ertr4d  8652  erinxp  8726  nqereq  10844  adderpq  10865  mulerpq  10866  efgred2  19680  efgcpbllemb  19682  efgcpbl2  19684  pcophtb  24983  pi1xfr  25009  pi1xfrcnvlem  25010  erbr3b  32644  prjspner1  42811  chnerlem1  47068