Theorem caovcand 6836

Description: Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovcang.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
caovcand.2	⊢ (𝜑 → 𝐴 ∈ 𝑇)
caovcand.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovcand.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)

Assertion

Ref	Expression
caovcand	⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

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

Proof of Theorem caovcand

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		caovcand.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
3		caovcand.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4		caovcand.4	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
5		caovcang.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
6	5	caovcang 6835	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
7	1, 2, 3, 4, 6	syl13anc 1328	1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  caovcanrd  6837