Theorem elimhyp 3710

Description: Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3703. (Contributed by NM, 15-May-1999.)

Hypotheses

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

Assertion

Ref	Expression
elimhyp	⊢ ψ

Proof of Theorem elimhyp

Step	Hyp	Ref	Expression
1		iftrue 3668	. . . . 5 ⊢ (φ → if(φ, A, B) = A)
2	1	eqcomd 2358	. . . 4 ⊢ (φ → A = if(φ, A, B))
3		elimhyp.1	. . . 4 ⊢ (A = if(φ, A, B) → (φ ↔ ψ))
4	2, 3	syl 15	. . 3 ⊢ (φ → (φ ↔ ψ))
5	4	ibi 232	. 2 ⊢ (φ → ψ)
6		elimhyp.3	. . 3 ⊢ χ
7		iffalse 3669	. . . . 5 ⊢ (¬ φ → if(φ, A, B) = B)
8	7	eqcomd 2358	. . . 4 ⊢ (¬ φ → B = if(φ, A, B))
9		elimhyp.2	. . . 4 ⊢ (B = if(φ, A, B) → (χ ↔ ψ))
10	8, 9	syl 15	. . 3 ⊢ (¬ φ → (χ ↔ ψ))
11	6, 10	mpbii 202	. 2 ⊢ (¬ φ → ψ)
12	5, 11	pm2.61i 156	1 ⊢ ψ

Syntax hints:  ¬ wn 3   → 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:  elimel  3714  elimf  5222