Theorem keephyp 3716

Description: Transform a hypothesis ψ that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
keephyp.1	⊢ (A = if(φ, A, B) → (ψ ↔ θ))
keephyp.2	⊢ (B = if(φ, A, B) → (χ ↔ θ))
keephyp.3	⊢ ψ
keephyp.4	⊢ χ

Assertion

Ref	Expression
keephyp	⊢ θ

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2 ⊢ ψ
2		keephyp.4	. 2 ⊢ χ
3		keephyp.1	. . 3 ⊢ (A = if(φ, A, B) → (ψ ↔ θ))
4		keephyp.2	. . 3 ⊢ (B = if(φ, A, B) → (χ ↔ θ))
5	3, 4	ifboth 3693	. 2 ⊢ ((ψ ∧ χ) → θ)
6	1, 2, 5	mp2an 653	1 ⊢ θ

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662

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 3663

This theorem is referenced by:  keepel  3719