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Metamath Proof Explorer

Theorem xrneq12d

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021)

Ref Expression
Hypotheses xrneq12d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
xrneq12d.2 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion xrneq12d ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 xrneq12d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 xrneq12d.2 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
3 xrneq12 ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ⋉ 𝐶 ) = ( 𝐵 ⋉ 𝐷 ) )