Theorem xrneq12d 39316

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 39314	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⋉ cxrn 39086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-in 3906  df-ss 3916  df-br 5104  df-opab 5168  df-co 5660  df-xrn 39292

This theorem is used by: (None)