Theorem xrneq12 39109

Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)

Assertion

Ref	Expression
xrneq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

Proof of Theorem xrneq12

Step	Hyp	Ref	Expression
1		xrneq1 39103	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
2		xrneq2 39106	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
3	1, 2	sylan9eq 2820	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⋉ cxrn 38881

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-in 3913  df-ss 3923  df-br 5112  df-opab 5176  df-co 5672  df-xrn 39087

This theorem is used by:  xrneq12i  39110  xrneq12d  39111