Theorem xrneq2i 37240

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 37239	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)

Syntax hints:   = wceq 1542   ⋉ cxrn 37031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3955  df-ss 3965  df-br 5149  df-opab 5211  df-co 5685  df-xrn 37230

This theorem is referenced by:  disjsuc  37618