Theorem xrneq2i 38367

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 38366	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)

Syntax hints:   = wceq 1540   ⋉ cxrn 38168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-in 3921  df-ss 3931  df-br 5108  df-opab 5170  df-co 5647  df-xrn 38353

This theorem is referenced by:  disjsuc  38751