Theorem xrneq1d 38284

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 38282	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))

Syntax hints:   → wi 4   = wceq 1537   ⋉ cxrn 38083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-in 3977  df-ss 3987  df-br 5170  df-opab 5232  df-co 5708  df-xrn 38276

This theorem is referenced by: (None)