Theorem ertrd 7758

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 7757	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
5	1, 2, 4	mp2and 715	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   class class class wbr 4653   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123  df-er 7742

This theorem is referenced by:  ertr2d  7759  ertr3d  7760  ertr4d  7761  erinxp  7821  nqereq  9757  adderpq  9778  mulerpq  9779  efgred2  18166  efgcpbllemb  18168  efgcpbl2  18170  pcophtb  22829  pi1xfr  22855  pi1xfrcnvlem  22856  erbr3b  29427