Theorem elimhyp 4504

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4497. (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 4445	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2743	. . . 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 270	. 2 ⊢ (𝜑 → 𝜓)
6		elimhyp.3	. . 3 ⊢ 𝜒
7		iffalse 4448	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2743	. . . 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 236	. 2 ⊢ (¬ 𝜑 → 𝜓)
12	5, 11	pm2.61i 185	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1543  ifcif 4439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4440

This theorem is referenced by:  elimel  4508  elimf  6544  oeoa  8325  oeoe  8327  limensuc  8823  axcc4dom  10055  elimne0  10823  elimgt0  11670  elimge0  11671  2ndcdisj  22353  siilem2  28933  normlem7tALT  29200  hhsssh  29350  shintcl  29411  chintcl  29413  spanun  29626  elunop2  30094  lnophm  30100  nmbdfnlb  30131  hmopidmch  30234  hmopidmpj  30235  chirred  30476  limsucncmp  34372  elimhyps  36712  elimhyps2  36715