Theorem caovord2 5673

Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	⊢ 𝐴 ∈ V
caovord.2	⊢ 𝐵 ∈ V
caovord.3	⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovord2.3	⊢ 𝐶 ∈ V
caovord2.com	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)

Assertion

Ref	Expression
caovord2	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

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

Proof of Theorem caovord2

Step	Hyp	Ref	Expression
1		caovord.1	. . 3 ⊢ 𝐴 ∈ V
2		caovord.2	. . 3 ⊢ 𝐵 ∈ V
3		caovord.3	. . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
4	1, 2, 3	caovord 5672	. 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5		caovord2.3	. . . 4 ⊢ 𝐶 ∈ V
6		caovord2.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
7	5, 1, 6	caovcom 5658	. . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
8	5, 2, 6	caovcom 5658	. . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
9	7, 8	breq12i 3773	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
10	4, 9	syl6bb 185	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  caovord3  5674