Theorem iffalse 3359

Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Assertion

Ref	Expression
iffalse	⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Proof of Theorem iffalse

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedlemb 911	. . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐵 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))))
2	1	abbi2dv 2197	. 2 ⊢ (¬ 𝜑 → 𝐵 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))})
3		df-if 3352	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
4	2, 3	syl6reqr 2132	1 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661   = wceq 1284   ∈ wcel 1433  {cab 2067  ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-if 3352

This theorem is referenced by:  iffalsei  3360  iffalsed  3361  ifnefalse  3362  ifsbdc  3363  ifcldadc  3378  ifeq1dadc  3379  ifbothdc  3380  ifcldcd  3381  fidifsnen  6355  uzin  8651  modifeq2int  9388  expival  9478  bcval  9676  bcval3  9678  flodddiv4  10334  gcdn0val  10353  dfgcd2  10403  lcmn0val  10448