Theorem xrneq2d 38897

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 38895	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))

Syntax hints:   → wi 4   = wceq 1560   ⋉ cxrn 38670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-in 3911  df-ss 3921  df-br 5101  df-opab 5163  df-co 5656  df-xrn 38876

This theorem is referenced by: (None)