Theorem caovordid 7629

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 7628	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146   class class class wbr 5114  (class class class)co 7423

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426

This theorem is used by: (None)