Theorem elimhyp 4532

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4525. (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 4472	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2742	. . . 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 267	. 2 ⊢ (𝜑 → 𝜓)
6		elimhyp.3	. . 3 ⊢ 𝜒
7		iffalse 4475	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2742	. . . 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 233	. 2 ⊢ (¬ 𝜑 → 𝜓)
12	5, 11	pm2.61i 182	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542  ifcif 4466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-if 4467

This theorem is referenced by:  elimel  4536  elimf  6667  oeoa  8533  oeoe  8535  limensuc  9092  axcc4dom  10363  elimne0  11134  elimgt0  11993  elimge0  11994  2ndcdisj  23421  siilem2  30923  normlem7tALT  31190  hhsssh  31340  shintcl  31401  chintcl  31403  spanun  31616  elunop2  32084  lnophm  32090  nmbdfnlb  32121  hmopidmch  32224  hmopidmpj  32225  chirred  32466  limsucncmp  36628  elimhyps  39407  elimhyps2  39410