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Metamath Proof Explorer

Theorem xrneq1i

Description: Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020)

Ref Expression
Hypothesis xrneq1i.1 ⊢ 𝐴 = 𝐵
Assertion xrneq1i ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 )

Proof

Step Hyp Ref Expression
1 xrneq1i.1 ⊢ 𝐴 = 𝐵
2 xrneq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 ) )
3 1 2 ax-mp ⊢ ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐶 )