Metamath Proof Explorer


Theorem ovmpordx

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf . (Contributed by AV, 30-Mar-2019)

Ref Expression
Hypotheses ovmpordx.1 ( 𝜑𝐹 = ( 𝑥𝐶 , 𝑦𝐷𝑅 ) )
ovmpordx.2 ( ( 𝜑 ∧ ( 𝑥 = 𝐴𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
ovmpordx.3 ( ( 𝜑𝑦 = 𝐵 ) → 𝐶 = 𝐿 )
ovmpordx.4 ( 𝜑𝐴𝐿 )
ovmpordx.5 ( 𝜑𝐵𝐷 )
ovmpordx.6 ( 𝜑𝑆𝑋 )
Assertion ovmpordx ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )

Proof

Step Hyp Ref Expression
1 ovmpordx.1 ( 𝜑𝐹 = ( 𝑥𝐶 , 𝑦𝐷𝑅 ) )
2 ovmpordx.2 ( ( 𝜑 ∧ ( 𝑥 = 𝐴𝑦 = 𝐵 ) ) → 𝑅 = 𝑆 )
3 ovmpordx.3 ( ( 𝜑𝑦 = 𝐵 ) → 𝐶 = 𝐿 )
4 ovmpordx.4 ( 𝜑𝐴𝐿 )
5 ovmpordx.5 ( 𝜑𝐵𝐷 )
6 ovmpordx.6 ( 𝜑𝑆𝑋 )
7 nfv 𝑥 𝜑
8 nfv 𝑦 𝜑
9 nfcv 𝑦 𝐴
10 nfcv 𝑥 𝐵
11 nfcv 𝑥 𝑆
12 nfcv 𝑦 𝑆
13 1 2 3 4 5 6 7 8 9 10 11 12 ovmpordxf ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = 𝑆 )