Theorem xrneq1i 35624

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

Hypothesis

Ref	Expression
xrneq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
xrneq1i	⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)

Proof of Theorem xrneq1i

Step	Hyp	Ref	Expression
1		xrneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		xrneq1 35623	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)

Syntax hints:   = wceq 1533   ⋉ cxrn 35446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-in 3942  df-ss 3951  df-br 5059  df-opab 5121  df-co 5558  df-xrn 35617

This theorem is referenced by: (None)