Theorem ifclda 3690

Description: Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifclda.1	⊢ ((φ ∧ ψ) → A ∈ C)
ifclda.2	⊢ ((φ ∧ ¬ ψ) → B ∈ C)

Assertion

Ref	Expression
ifclda	⊢ (φ → if(ψ, A, B) ∈ C)

Proof of Theorem ifclda

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . 4 ⊢ (ψ → if(ψ, A, B) = A)
2	1	adantl 452	. . 3 ⊢ ((φ ∧ ψ) → if(ψ, A, B) = A)
3		ifclda.1	. . 3 ⊢ ((φ ∧ ψ) → A ∈ C)
4	2, 3	eqeltrd 2427	. 2 ⊢ ((φ ∧ ψ) → if(ψ, A, B) ∈ C)
5		iffalse 3670	. . . 4 ⊢ (¬ ψ → if(ψ, A, B) = B)
6	5	adantl 452	. . 3 ⊢ ((φ ∧ ¬ ψ) → if(ψ, A, B) = B)
7		ifclda.2	. . 3 ⊢ ((φ ∧ ¬ ψ) → B ∈ C)
8	6, 7	eqeltrd 2427	. 2 ⊢ ((φ ∧ ¬ ψ) → if(ψ, A, B) ∈ C)
9	4, 8	pm2.61dan 766	1 ⊢ (φ → if(ψ, A, B) ∈ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   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: (None)