Theorem ifeq12 4500

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 4485	. 2 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2		ifeq2 4486	. 2 ⊢ (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
3	1, 2	sylan9eq 2815	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ifcif 4481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 4150  df-if 4482

This theorem is used by:  xaddmnf1  13365  xpsrnbas  17752  ditg0  26182  mumullem2  27516  sqrtcval  44778  sqrtcval2  44779