Theorem caovord3d 7652

Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovordg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovordd.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovordd.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovordd.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovord2d.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovord3d.5	⊢ (𝜑 → 𝐷 ∈ 𝑆)

Assertion

Ref	Expression
caovord3d	⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)))

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

Proof of Theorem caovord3d

Step	Hyp	Ref	Expression
1		breq1 5109	. 2 ⊢ ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
2		caovordg.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
3		caovordd.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
4		caovordd.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
5		caovordd.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
6		caovord2d.com	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
7	2, 3, 4, 5, 6	caovord2d 7651	. . 3 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
8		caovord3d.5	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
9	2, 8, 5, 4	caovordd 7650	. . 3 ⊢ (𝜑 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
10	7, 9	bibi12d 348	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵) ↔ ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))))
11	1, 10	imbitrrid 249	1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by: (None)