Theorem iffalsei 4492

Description: Inference associated with iffalse 4491. (Contributed by BJ, 7-Oct-2018.)

Hypothesis

Ref	Expression
iffalsei.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
iffalsei	⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

Proof of Theorem iffalsei

Step	Hyp	Ref	Expression
1		iffalsei.1	. 2 ⊢ ¬ 𝜑
2		iffalse 4491	. 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	1, 2	ax-mp 5	1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4483

This theorem is used by:  ssttrcl  9709  ttrclselem2  9720  sum0  15880  prod0  16103  prmo4  17299  prmo6  17301  itg0  26093  vieta1lem2  26627  right1s  28275  vtxval0  29610  iedgval0  29611  ex-prmo  31053  dfrdg2  36537  dfrdg4  36695  fwddifnp1  36910  bj-pr21val  37906  bj-pr22val  37912  imsqrtvalex  44631  clsk1indlem4  45029  clsk1indlem1  45030  refsum2cnlem1  46023  limsup10ex  46752  iblempty  46944  fouriersw  47210