Theorem ifeq12 3344

Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)

Assertion

Ref	Expression
ifeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Proof of Theorem ifeq12

Step	Hyp	Ref	Expression
1		ifeq1 3334	. 2 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2		ifeq2 3335	. 2 ⊢ (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
3	1, 2	sylan9eq 2092	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-if 3332

This theorem is referenced by: (None)