Metamath Proof Explorer


Theorem xrneq1d

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021)

Ref Expression
Hypothesis xrneq1d.1 ( 𝜑𝐴 = 𝐵 )
Assertion xrneq1d ( 𝜑 → ( 𝐴𝐶 ) = ( 𝐵𝐶 ) )

Proof

Step Hyp Ref Expression
1 xrneq1d.1 ( 𝜑𝐴 = 𝐵 )
2 xrneq1 ( 𝐴 = 𝐵 → ( 𝐴𝐶 ) = ( 𝐵𝐶 ) )
3 1 2 syl ( 𝜑 → ( 𝐴𝐶 ) = ( 𝐵𝐶 ) )