Theorem elimhyp 4523

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4516. (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 4466	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2826	. . . 4 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵))
3		elimhyp.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (𝜑 ↔ 𝜓))
5	4	ibi 269	. 2 ⊢ (𝜑 → 𝜓)
6		elimhyp.3	. . 3 ⊢ 𝜒
7		iffalse 4469	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2826	. . . 4 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵))
9		elimhyp.2	. . . 4 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓))
10	8, 9	syl 17	. . 3 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜓))
11	6, 10	mpbii 235	. 2 ⊢ (¬ 𝜑 → 𝜓)
12	5, 11	pm2.61i 184	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1536  ifcif 4460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-if 4461

This theorem is referenced by:  elimel  4527  elimf  6506  oeoa  8216  oeoe  8218  limensuc  8687  axcc4dom  9856  elimne0  10624  elimgt0  11471  elimge0  11472  2ndcdisj  22059  siilem2  28627  normlem7tALT  28894  hhsssh  29044  shintcl  29105  chintcl  29107  spanun  29320  elunop2  29788  lnophm  29794  nmbdfnlb  29825  hmopidmch  29928  hmopidmpj  29929  chirred  30170  limsucncmp  33815  elimhyps  36130  elimhyps2  36133