Theorem xrneq1d 38375

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

Hypothesis

Ref	Expression
xrneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
xrneq1d	⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Proof of Theorem xrneq1d

Step	Hyp	Ref	Expression
1		xrneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		xrneq1 38373	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Syntax hints:   → wi 4   = wceq 1539   ⋉ cxrn 38175

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-in 3973  df-ss 3983  df-br 5152  df-opab 5214  df-co 5702  df-xrn 38367

This theorem is referenced by: (None)