Theorem caovordid 7618

Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypotheses

Ref	Expression
caovordig.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovordid.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovordid.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovordid.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)

Assertion

Ref	Expression
caovordid	⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

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

Proof of Theorem caovordid

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝜑 → 𝜑)
2		caovordid.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovordid.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4		caovordid.4	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
5		caovordig.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
6	5	caovordig 7617	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
7	1, 2, 3, 4, 6	syl13anc 1399	1 ⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143   class class class wbr 5110  (class class class)co 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415

This theorem is referenced by: (None)