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Theorem ifbothda 3692
 Description: A wff θ containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
Hypotheses
Ref Expression
ifboth.1 (A = if(φ, A, B) → (ψθ))
ifboth.2 (B = if(φ, A, B) → (χθ))
ifbothda.3 ((η φ) → ψ)
ifbothda.4 ((η ¬ φ) → χ)
Assertion
Ref Expression
ifbothda (ηθ)

Proof of Theorem ifbothda
StepHypRef Expression
1 ifbothda.3 . . 3 ((η φ) → ψ)
2 iftrue 3668 . . . . . 6 (φ → if(φ, A, B) = A)
32eqcomd 2358 . . . . 5 (φA = if(φ, A, B))
4 ifboth.1 . . . . 5 (A = if(φ, A, B) → (ψθ))
53, 4syl 15 . . . 4 (φ → (ψθ))
65adantl 452 . . 3 ((η φ) → (ψθ))
71, 6mpbid 201 . 2 ((η φ) → θ)
8 ifbothda.4 . . 3 ((η ¬ φ) → χ)
9 iffalse 3669 . . . . . 6 φ → if(φ, A, B) = B)
109eqcomd 2358 . . . . 5 φB = if(φ, A, B))
11 ifboth.2 . . . . 5 (B = if(φ, A, B) → (χθ))
1210, 11syl 15 . . . 4 φ → (χθ))
1312adantl 452 . . 3 ((η ¬ φ) → (χθ))
148, 13mpbid 201 . 2 ((η ¬ φ) → θ)
157, 14pm2.61dan 766 1 (ηθ)
 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:  ifboth  3693
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