Theorem ertr3d 8692

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ertr3d.5	⊢ (𝜑 → 𝐵𝑅𝐴)
ertr3d.6	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
ertr3d	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem ertr3d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ertr3d.5	. . 3 ⊢ (𝜑 → 𝐵𝑅𝐴)
3	1, 2	ersym 8686	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
4		ertr3d.6	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 3, 4	ertrd 8690	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5110   Er wer 8671

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-er 8674

This theorem is referenced by:  nqereq  10895  efgred2  19690  xmetresbl  24332  pcophtb  24936  pi1xfr  24962  pi1xfrcnvlem  24963  erbr3b  32552  prtlem10  38865