Theorem elimhyp 4540

Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4533. (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 4480	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2739	. . . 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 4483	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2739	. . . 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 1541  ifcif 4474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-if 4475

This theorem is referenced by:  elimel  4544  elimf  6655  oeoa  8518  oeoe  8520  limensuc  9074  axcc4dom  10339  elimne0  11109  elimgt0  11966  elimge0  11967  2ndcdisj  23372  siilem2  30834  normlem7tALT  31101  hhsssh  31251  shintcl  31312  chintcl  31314  spanun  31527  elunop2  31995  lnophm  32001  nmbdfnlb  32032  hmopidmch  32135  hmopidmpj  32136  chirred  32377  limsucncmp  36511  elimhyps  39081  elimhyps2  39084