Theorem caovcanrd 6217

Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovcang.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
caovcand.2	⊢ (𝜑 → 𝐴 ∈ 𝑇)
caovcand.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovcand.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovcanrd.5	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovcanrd.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))

Assertion

Ref	Expression
caovcanrd	⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧

Proof of Theorem caovcanrd

Step	Hyp	Ref	Expression
1		caovcanrd.6	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2		caovcanrd.5	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovcand.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4	1, 2, 3	caovcomd 6210	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
5		caovcand.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
6	1, 2, 5	caovcomd 6210	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐶𝐹𝐴))
7	4, 6	eqeq12d 2247	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ (𝐵𝐹𝐴) = (𝐶𝐹𝐴)))
8		caovcang.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
9		caovcand.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
10	8, 9, 3, 5	caovcand 6216	. 2 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
11	7, 10	bitr3d 190	1 ⊢ (𝜑 → ((𝐵𝐹𝐴) = (𝐶𝐹𝐴) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  (class class class)co 6049

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-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by: (None)