Theorem ovif2 7255

Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.)

Assertion

Ref	Expression
ovif2	⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Proof of Theorem ovif2

Step	Hyp	Ref	Expression
1		oveq2 7167	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2		oveq2 7167	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
3	1, 2	ifsb 4483	1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Syntax hints:   = wceq 1536  ifcif 4470  (class class class)co 7159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-ov 7162

This theorem is referenced by:  ramcl  16368  matsc  21062  scmatscmide  21119  mulmarep1el  21184  maducoeval2  21252  madugsum  21255  itg2const  24344  itg2monolem1  24354  iblmulc2  24434  itgmulc2lem1  24435  bddmulibl  24442  dchrvmasumiflem2  26081  rpvmasum2  26091  sgnneg  31802  itg2addnclem  34947  itgaddnclem2  34955  itgmulc2nclem1  34962