Theorem ifeq12 4547

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 4533	. 2 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
2		ifeq2 4534	. 2 ⊢ (𝐶 = 𝐷 → if(𝜑, 𝐵, 𝐶) = if(𝜑, 𝐵, 𝐷))
3	1, 2	sylan9eq 2793	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ifcif 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-un 3954  df-if 4530

This theorem is referenced by:  xaddmnf1  13207  xpsrnbas  17517  ditg0  25370  mumullem2  26684  sqrtcval  42392  sqrtcval2  42393