Theorem cnveqb 6218

Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
cnveqb	⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))

Proof of Theorem cnveqb

Step	Hyp	Ref	Expression
1		cnveq 5887	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
2		dfrel2 6211	. . . 4 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
3		dfrel2 6211	. . . . . . 7 ⊢ (Rel 𝐵 ↔ ◡◡𝐵 = 𝐵)
4		cnveq 5887	. . . . . . . . 9 ⊢ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = ◡◡𝐵)
5		eqeq2 2747	. . . . . . . . 9 ⊢ (𝐵 = ◡◡𝐵 → (◡◡𝐴 = 𝐵 ↔ ◡◡𝐴 = ◡◡𝐵))
6	4, 5	imbitrrid 246	. . . . . . . 8 ⊢ (𝐵 = ◡◡𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
7	6	eqcoms 2743	. . . . . . 7 ⊢ (◡◡𝐵 = 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
8	3, 7	sylbi 217	. . . . . 6 ⊢ (Rel 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
9		eqeq1 2739	. . . . . . 7 ⊢ (𝐴 = ◡◡𝐴 → (𝐴 = 𝐵 ↔ ◡◡𝐴 = 𝐵))
10	9	imbi2d 340	. . . . . 6 ⊢ (𝐴 = ◡◡𝐴 → ((◡𝐴 = ◡𝐵 → 𝐴 = 𝐵) ↔ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)))
11	8, 10	imbitrrid 246	. . . . 5 ⊢ (𝐴 = ◡◡𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
12	11	eqcoms 2743	. . . 4 ⊢ (◡◡𝐴 = 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
13	2, 12	sylbi 217	. . 3 ⊢ (Rel 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
14	13	imp 406	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))
15	1, 14	impbid2 226	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ^◡ccnv 5688  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697

This theorem is referenced by:  cnveq0  6219  weisoeq2  7376  relexpaddg  15089  relexpaddss  43708