Theorem ifboth 3694

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) → (ψ ↔ θ))
ifboth.2	⊢ (B = if(φ, A, B) → (χ ↔ θ))

Assertion

Ref	Expression
ifboth	⊢ ((ψ ∧ χ) → θ)

Proof of Theorem ifboth

Step	Hyp	Ref	Expression
1		ifboth.1	. 2 ⊢ (A = if(φ, A, B) → (ψ ↔ θ))
2		ifboth.2	. 2 ⊢ (B = if(φ, A, B) → (χ ↔ θ))
3		simpll 730	. 2 ⊢ (((ψ ∧ χ) ∧ φ) → ψ)
4		simplr 731	. 2 ⊢ (((ψ ∧ χ) ∧ ¬ φ) → χ)
5	1, 2, 3, 4	ifbothda 3693	1 ⊢ ((ψ ∧ χ) → θ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 3664

This theorem is referenced by:  ifcl  3699  keephyp  3717