Theorem xrneq12d 38652

Description: Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)

Hypotheses

Ref	Expression
xrneq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
xrneq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
xrneq12d	⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

Proof of Theorem xrneq12d

Step	Hyp	Ref	Expression
1		xrneq12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		xrneq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		xrneq12 38650	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

Syntax hints:   → wi 4   = wceq 1542   ⋉ cxrn 38422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-in 3910  df-ss 3920  df-br 5101  df-opab 5163  df-co 5641  df-xrn 38628

This theorem is referenced by: (None)