Theorem erth2 6474

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 6443	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
3		erth2.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4	1, 3	erth 6473	. . 3 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅))
5		eqcom 2141	. . 3 ⊢ ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅)
6	4, 5	syl6bb 195	. 2 ⊢ (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅))
7	2, 6	bitrd 187	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480   class class class wbr 3929   Er wer 6426  [cec 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-er 6429  df-ec 6431

This theorem is referenced by:  qliftel  6509