Theorem keephyp 1794

Description: Transform a hypothesis ψ that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem.

Hypotheses

Ref	Expression
keephyp.1	⊢ (A = if(φ, A, B) → (ψ ↔ θ))
keephyp.2	⊢ (B = if(φ, A, B) → (χ ↔ θ))
keephyp.3	⊢ ψ
keephyp.4	⊢ χ

Assertion

Ref	Expression
keephyp	⊢ θ

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. . 3 ⊢ ψ
2		iftrue 1780	. . . . 5 ⊢ (φ → if(φ, A, B) = A)
3	2	cleqcomd 1106	. . . 4 ⊢ (φ → A = if(φ, A, B))
4		keephyp.1	. . . 4 ⊢ (A = if(φ, A, B) → (ψ ↔ θ))
5	3, 4	syl 12	. . 3 ⊢ (φ → (ψ ↔ θ))
6	1, 5	mpbii 168	. 2 ⊢ (φ → θ)
7		keephyp.4	. . 3 ⊢ χ
8		iffalse 1781	. . . . 5 ⊢ (¬ φ → if(φ, A, B) = B)
9	8	cleqcomd 1106	. . . 4 ⊢ (¬ φ → B = if(φ, A, B))
10		keephyp.2	. . . 4 ⊢ (B = if(φ, A, B) → (χ ↔ θ))
11	9, 10	syl 12	. . 3 ⊢ (¬ φ → (χ ↔ θ))
12	7, 11	mpbii 168	. 2 ⊢ (¬ φ → θ)
13	6, 12	pm2.61i 110	1 ⊢ θ

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

This theorem is referenced by:  keepel 1796  mulcant 4208  sqrlem21 4751  sqrlem22 4752  projlem7 5199

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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