Theorem dedth4h 1789

Description: Weak deduction theorem eliminating four hypotheses.

Hypotheses

Ref	Expression
dedth4h.1	⊢ (A = if(φ, A, R) → (τ ↔ η))
dedth4h.2	⊢ (B = if(ψ, B, S) → (η ↔ ζ))
dedth4h.3	⊢ (C = if(χ, C, F) → (ζ ↔ σ))
dedth4h.4	⊢ (D = if(θ, D, G) → (σ ↔ ρ))
dedth4h.5	⊢ ρ

Assertion

Ref	Expression
dedth4h	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)

Proof of Theorem dedth4h

Step	Hyp	Ref	Expression
1		dedth4h.1	. . . 4 ⊢ (A = if(φ, A, R) → (τ ↔ η))
2	1	imbi2d 464	. . 3 ⊢ (A = if(φ, A, R) → (((χ ∧ θ) → τ) ↔ ((χ ∧ θ) → η)))
3		dedth4h.2	. . . 4 ⊢ (B = if(ψ, B, S) → (η ↔ ζ))
4	3	imbi2d 464	. . 3 ⊢ (B = if(ψ, B, S) → (((χ ∧ θ) → η) ↔ ((χ ∧ θ) → ζ)))
5		dedth4h.3	. . . 4 ⊢ (C = if(χ, C, F) → (ζ ↔ σ))
6		dedth4h.4	. . . 4 ⊢ (D = if(θ, D, G) → (σ ↔ ρ))
7		dedth4h.5	. . . 4 ⊢ ρ
8	5, 6, 7	dedth2h 1787	. . 3 ⊢ ((χ ∧ θ) → ζ)
9	2, 4, 8	dedth2h 1787	. 2 ⊢ ((φ ∧ ψ) → ((χ ∧ θ) → τ))
10	9	imp 277	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ifcif 1776

This theorem is referenced by:  lt2addt 4361  crut 4531  nn0opth2t 4726  abs3lemt 4865  hvsubsub4t 5032  norm3lemt 5097  projlem20 5212

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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