Theorem ertr3d 6619

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 6613	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
4		ertr3d.6	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
5	1, 3, 4	ertrd 6617	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 4034   Er wer 6598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-er 6601

This theorem is referenced by:  xmetresbl  14760