Theorem caovcom 6076

Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Hypotheses

Ref	Expression
caovcom.1	⊢ 𝐴 ∈ V
caovcom.2	⊢ 𝐵 ∈ V
caovcom.3	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)

Assertion

Ref	Expression
caovcom	⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem caovcom

Step	Hyp	Ref	Expression
1		caovcom.1	. 2 ⊢ 𝐴 ∈ V
2		caovcom.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	pm3.2i 272	. 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)
4		caovcom.3	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5	4	a1i 9	. . 3 ⊢ ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
6	5	caovcomg 6074	. 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
7	1, 3, 6	mp2an 426	1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  caovord2  6091  caov32  6106  caov12  6107  ecopovsym  6685  ecopover  6687