Theorem caovord3 7616

Description: Ordering law. (Contributed by NM, 29-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)
caovord3.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
caovord3	⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))

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

Proof of Theorem caovord3

Step	Hyp	Ref	Expression
1		caovord.1	. . . . 5 ⊢ 𝐴 ∈ V
2		caovord2.3	. . . . 5 ⊢ 𝐶 ∈ V
3		caovord.3	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
4		caovord.2	. . . . 5 ⊢ 𝐵 ∈ V
5		caovord2.com	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
6	1, 2, 3, 4, 5	caovord2 7615	. . . 4 ⊢ (𝐵 ∈ 𝑆 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
7	6	adantr 480	. . 3 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
8		breq1 5144	. . 3 ⊢ ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
9	7, 8	sylan9bb 509	. 2 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
10		caovord3.4	. . . 4 ⊢ 𝐷 ∈ V
11	10, 4, 3	caovord 7614	. . 3 ⊢ (𝐶 ∈ 𝑆 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
12	11	ad2antlr 724	. 2 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
13	9, 12	bitr4d 282	1 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  (class class class)co 7404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407

This theorem is referenced by:  genpnnp  10999  ltsrpr  11071