Theorem xrneq2d 38768

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

Hypothesis

Ref	Expression
xrneq2d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
xrneq2d	⊢ (𝜑 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Proof of Theorem xrneq2d

Step	Hyp	Ref	Expression
1		xrneq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		xrneq2 38766	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1547   ⋉ cxrn 38541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-in 3890  df-ss 3900  df-br 5073  df-opab 5135  df-co 5627  df-xrn 38747

This theorem is referenced by: (None)