Theorem dedth3v 1786

Description: Weak deduction theorem for eliminating hypothesis with 3 class variables.

Hypotheses

Ref	Expression
dedth3v.1	⊢ (A = if(φ, A, D) → (ψ ↔ χ))
dedth3v.2	⊢ (B = if(φ, B, R) → (χ ↔ θ))
dedth3v.3	⊢ (C = if(φ, C, S) → (θ ↔ τ))
dedth3v.4	⊢ τ

Assertion

Ref	Expression
dedth3v	⊢ (φ → ψ)

Proof of Theorem dedth3v

Step	Hyp	Ref	Expression
1		dedth3v.4	. 2 ⊢ τ
2		iftrue 1780	. . . . 5 ⊢ (φ → if(φ, A, D) = A)
3	2	cleqcomd 1106	. . . 4 ⊢ (φ → A = if(φ, A, D))
4		dedth3v.1	. . . 4 ⊢ (A = if(φ, A, D) → (ψ ↔ χ))
5	3, 4	syl 12	. . 3 ⊢ (φ → (ψ ↔ χ))
6		iftrue 1780	. . . . 5 ⊢ (φ → if(φ, B, R) = B)
7	6	cleqcomd 1106	. . . 4 ⊢ (φ → B = if(φ, B, R))
8		dedth3v.2	. . . 4 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
9	7, 8	syl 12	. . 3 ⊢ (φ → (χ ↔ θ))
10		iftrue 1780	. . . . 5 ⊢ (φ → if(φ, C, S) = C)
11	10	cleqcomd 1106	. . . 4 ⊢ (φ → C = if(φ, C, S))
12		dedth3v.3	. . . 4 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
13	11, 12	syl 12	. . 3 ⊢ (φ → (θ ↔ τ))
14	5, 9, 13	3bitrd 422	. 2 ⊢ (φ → (ψ ↔ τ))
15	1, 14	mpbiri 169	1 ⊢ (φ → ψ)

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

This theorem is referenced by:  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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