Theorem elimhyp 4551

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4544. (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 4491	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2769	. . . 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 4494	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2769	. . . 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

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-if 4486

This theorem is used by:  elimel  4555  elimf  6730  oeoa  8634  oeoe  8636  limensuc  9201  axcc4dom  10546  elimne0  11323  elimgt0  12182  elimge0  12183  2ndcdisj  23805  siilem2  31484  normlem7tALT  31751  hhsssh  31901  shintcl  31962  chintcl  31964  spanun  32177  elunop2  32645  lnophm  32651  nmbdfnlb  32682  hmopidmch  32785  hmopidmpj  32786  chirred  33027  limsucncmp  37412  elimhyps  40210  elimhyps2  40213