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Theorem ifboth 3693
 Description: A wff θ containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
Hypotheses
Ref Expression
ifboth.1 (A = if(φ, A, B) → (ψθ))
ifboth.2 (B = if(φ, A, B) → (χθ))
Assertion
Ref Expression
ifboth ((ψ χ) → θ)

Proof of Theorem ifboth
StepHypRef Expression
1 ifboth.1 . 2 (A = if(φ, A, B) → (ψθ))
2 ifboth.2 . 2 (B = if(φ, A, B) → (χθ))
3 simpll 730 . 2 (((ψ χ) φ) → ψ)
4 simplr 731 . 2 (((ψ χ) ¬ φ) → χ)
51, 2, 3, 4ifbothda 3692 1 ((ψ χ) → θ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  ifcl  3698  keephyp  3716
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