Theorem elimhyp 1790

Description: Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.

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 1780	. . . . 5 ⊢ (φ → if(φ, A, B) = A)
2	1	cleqcomd 1106	. . . 4 ⊢ (φ → A = if(φ, A, B))
3		elimhyp.1	. . . 4 ⊢ (A = if(φ, A, B) → (φ ↔ ψ))
4	2, 3	syl 12	. . 3 ⊢ (φ → (φ ↔ ψ))
5	4	ibi 449	. 2 ⊢ (φ → ψ)
6		elimhyp.3	. . 3 ⊢ χ
7		iffalse 1781	. . . . 5 ⊢ (¬ φ → if(φ, A, B) = B)
8	7	cleqcomd 1106	. . . 4 ⊢ (¬ φ → B = if(φ, A, B))
9		elimhyp.2	. . . 4 ⊢ (B = if(φ, A, B) → (χ ↔ ψ))
10	8, 9	syl 12	. . 3 ⊢ (¬ φ → (χ ↔ ψ))
11	6, 10	mpbii 168	. 2 ⊢ (¬ φ → ψ)
12	5, 11	pm2.61i 110	1 ⊢ ψ

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091  ifcif 1776

This theorem is referenced by:  elimel 1793  limensuc 3402  elimne0 4102  dividt 4256  recrect 4260  elimgt0 4381  elimge0 4382  sqrlem20 4750  sqrlem21 4751  sqrlem22 4752  ruclem39 4923  normlem7t 5072  occlt 5189  shintclt 5295  chintclt 5297

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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