Theorem cnveqb 5199

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 4910	. 2 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
2		dfrel2 5194	. . . 4 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
3		dfrel2 5194	. . . . . . 7 ⊢ (Rel 𝐵 ↔ ◡◡𝐵 = 𝐵)
4		cnveq 4910	. . . . . . . . 9 ⊢ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = ◡◡𝐵)
5		eqeq2 2241	. . . . . . . . 9 ⊢ (𝐵 = ◡◡𝐵 → (◡◡𝐴 = 𝐵 ↔ ◡◡𝐴 = ◡◡𝐵))
6	4, 5	imbitrrid 156	. . . . . . . 8 ⊢ (𝐵 = ◡◡𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
7	6	eqcoms 2234	. . . . . . 7 ⊢ (◡◡𝐵 = 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
8	3, 7	sylbi 121	. . . . . 6 ⊢ (Rel 𝐵 → (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵))
9		eqeq1 2238	. . . . . . 7 ⊢ (𝐴 = ◡◡𝐴 → (𝐴 = 𝐵 ↔ ◡◡𝐴 = 𝐵))
10	9	imbi2d 230	. . . . . 6 ⊢ (𝐴 = ◡◡𝐴 → ((◡𝐴 = ◡𝐵 → 𝐴 = 𝐵) ↔ (◡𝐴 = ◡𝐵 → ◡◡𝐴 = 𝐵)))
11	8, 10	imbitrrid 156	. . . . 5 ⊢ (𝐴 = ◡◡𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
12	11	eqcoms 2234	. . . 4 ⊢ (◡◡𝐴 = 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
13	2, 12	sylbi 121	. . 3 ⊢ (Rel 𝐴 → (Rel 𝐵 → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵)))
14	13	imp 124	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (◡𝐴 = ◡𝐵 → 𝐴 = 𝐵))
15	1, 14	impbid2 143	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ^◡ccnv 4730  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739

This theorem is referenced by:  cnveq0  5200