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Theorem ertrd 8162
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
ertrd.5 (𝜑𝐴𝑅𝐵)
ertrd.6 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
ertrd (𝜑𝐴𝑅𝐶)

Proof of Theorem ertrd
StepHypRef Expression
1 ertrd.5 . 2 (𝜑𝐴𝑅𝐵)
2 ertrd.6 . 2 (𝜑𝐵𝑅𝐶)
3 ersymb.1 . . 3 (𝜑𝑅 Er 𝑋)
43ertr 8161 . 2 (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
51, 2, 4mp2and 695 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4968   Er wer 8143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-co 5459  df-er 8146
This theorem is referenced by:  ertr2d  8163  ertr3d  8164  ertr4d  8165  erinxp  8228  nqereq  10210  adderpq  10231  mulerpq  10232  efgred2  18610  efgcpbllemb  18612  efgcpbl2  18614  pcophtb  23320  pi1xfr  23346  pi1xfrcnvlem  23347  erbr3b  30054
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