Theorem ifbothda 3693

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) → (ψ ↔ θ))
ifboth.2	⊢ (B = if(φ, A, B) → (χ ↔ θ))
ifbothda.3	⊢ ((η ∧ φ) → ψ)
ifbothda.4	⊢ ((η ∧ ¬ φ) → χ)

Assertion

Ref	Expression
ifbothda	⊢ (η → θ)

Proof of Theorem ifbothda

Step	Hyp	Ref	Expression
1		ifbothda.3	. . 3 ⊢ ((η ∧ φ) → ψ)
2		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, A, B) = A)
3	2	eqcomd 2358	. . . . 5 ⊢ (φ → A = if(φ, A, B))
4		ifboth.1	. . . . 5 ⊢ (A = if(φ, A, B) → (ψ ↔ θ))
5	3, 4	syl 15	. . . 4 ⊢ (φ → (ψ ↔ θ))
6	5	adantl 452	. . 3 ⊢ ((η ∧ φ) → (ψ ↔ θ))
7	1, 6	mpbid 201	. 2 ⊢ ((η ∧ φ) → θ)
8		ifbothda.4	. . 3 ⊢ ((η ∧ ¬ φ) → χ)
9		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, A, B) = B)
10	9	eqcomd 2358	. . . . 5 ⊢ (¬ φ → B = if(φ, A, B))
11		ifboth.2	. . . . 5 ⊢ (B = if(φ, A, B) → (χ ↔ θ))
12	10, 11	syl 15	. . . 4 ⊢ (¬ φ → (χ ↔ θ))
13	12	adantl 452	. . 3 ⊢ ((η ∧ ¬ φ) → (χ ↔ θ))
14	8, 13	mpbid 201	. 2 ⊢ ((η ∧ ¬ φ) → θ)
15	7, 14	pm2.61dan 766	1 ⊢ (η → 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:  ifboth  3694