Theorem elimhyp 4527

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4520. (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 4467	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2746	. . . 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 268	. 2 ⊢ (𝜑 → 𝜓)
6		elimhyp.3	. . 3 ⊢ 𝜒
7		iffalse 4470	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2746	. . . 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 234	. 2 ⊢ (¬ 𝜑 → 𝜓)
12	5, 11	pm2.61i 183	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   = wceq 1547  ifcif 4461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-if 4462

This theorem is referenced by:  elimel  4531  elimf  6661  oeoa  8530  oeoe  8532  limensuc  9089  axcc4dom  10361  elimne0  11132  elimgt0  11991  elimge0  11992  2ndcdisj  23446  siilem2  30948  normlem7tALT  31215  hhsssh  31365  shintcl  31426  chintcl  31428  spanun  31641  elunop2  32109  lnophm  32115  nmbdfnlb  32146  hmopidmch  32249  hmopidmpj  32250  chirred  32491  limsucncmp  36681  elimhyps  39460  elimhyps2  39463