Theorem xrneq2i 38316

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

Hypothesis

Ref	Expression
xrneq2i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
xrneq2i	⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)

Proof of Theorem xrneq2i

Step	Hyp	Ref	Expression
1		xrneq2i.1	. 2 ⊢ 𝐴 = 𝐵
2		xrneq2 38315	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)

Syntax hints:   = wceq 1539   ⋉ cxrn 38115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-in 3938  df-ss 3948  df-br 5124  df-opab 5186  df-co 5674  df-xrn 38306

This theorem is referenced by:  disjsuc  38694