Theorem ertr4d 8517

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

Hypotheses

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)
ertr4d.5	⊢ (𝜑 → 𝐴𝑅𝐵)
ertr4d.6	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
ertr4d	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem ertr4d

Step	Hyp	Ref	Expression
1		ersymb.1	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		ertr4d.5	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		ertr4d.6	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
4	1, 3	ersym 8510	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 2, 4	ertrd 8514	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 5074   Er wer 8495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-er 8498

This theorem is referenced by:  erref  8518  erdisj  8550  nqereu  10685  nqereq  10691  efgredeu  19358  pi1xfr  24218  pi1xfrcnvlem  24219  prjspner1  40463