Theorem erth2 8506

Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
erth2.1	⊢ (𝜑 → 𝑅 Er 𝑋)
erth2.2	⊢ (𝜑 → 𝐵 ∈ 𝑋)

Assertion

Ref	Expression
erth2	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem erth2

Step	Hyp	Ref	Expression
1		erth2.1	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
2	1	ersymb 8470	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
3		erth2.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4	1, 3	erth 8505	. . 3 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅))
5		eqcom 2745	. . 3 ⊢ ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅)
6	4, 5	bitrdi 286	. 2 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅))
7	2, 6	bitrd 278	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108   class class class wbr 5070   Er wer 8453  [cec 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-er 8456  df-ec 8458

This theorem is referenced by:  qliftel  8547  qusker  31451