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Metamath Proof Explorer

Theorem caovcan

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995)

Ref Expression
Hypotheses caovcan.1 ⊢ 𝐶 ∈ V
caovcan.2 ⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) )
Assertion caovcan ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 caovcan.1 ⊢ 𝐶 ∈ V
2 caovcan.2 ⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) )
3 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) )
4 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝐶 ) = ( 𝐴 𝐹 𝐶 ) )
5 3 4 eqeq12d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ) )
6 5 imbi1d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) )
7 oveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
8 7 eqeq1d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ) )
9 eqeq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝐶 ↔ 𝐵 = 𝐶 ) )
10 8 9 imbi12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) )
11 oveq2 ⊢ ( 𝑧 = 𝐶 → ( 𝑥 𝐹 𝑧 ) = ( 𝑥 𝐹 𝐶 ) )
12 11 eqeq2d ⊢ ( 𝑧 = 𝐶 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ) )
13 eqeq2 ⊢ ( 𝑧 = 𝐶 → ( 𝑦 = 𝑧 ↔ 𝑦 = 𝐶 ) )
14 12 13 imbi12d ⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) )
15 14 imbi2d ⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ) ↔ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) ) )
16 1 15 2 vtocl ⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) )
17 6 10 16 vtocl2ga ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )