Theorem elimhyp 4549

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4542. (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 4489	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2771	. . . 4 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵))
3		elimhyp.1	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓))
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → (𝜑 ↔ 𝜓))
5	4	ibi 270	. 2 ⊢ (𝜑 → 𝜓)
6		elimhyp.3	. . 3 ⊢ 𝜒
7		iffalse 4492	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2771	. . . 4 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵))
9		elimhyp.2	. . . 4 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓))
10	8, 9	syl 18	. . 3 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜓))
11	6, 10	mpbii 236	. 2 ⊢ (¬ 𝜑 → 𝜓)
12	5, 11	pm2.61i 184	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1563  ifcif 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-if 4484

This theorem is referenced by:  elimel  4553  elimf  6694  oeoa  8571  oeoe  8573  limensuc  9130  axcc4dom  10413  elimne0  11184  elimgt0  12044  elimge0  12045  2ndcdisj  23574  siilem2  31113  normlem7tALT  31380  hhsssh  31530  shintcl  31591  chintcl  31593  spanun  31806  elunop2  32274  lnophm  32280  nmbdfnlb  32311  hmopidmch  32414  hmopidmpj  32415  chirred  32656  limsucncmp  36819  elimhyps  39597  elimhyps2  39600