Theorem syl3anbr 1226

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)

Hypotheses

Ref	Expression
syl3anbr.1	⊢ (ψ ↔ φ)
syl3anbr.2	⊢ (θ ↔ χ)
syl3anbr.3	⊢ (η ↔ τ)
syl3anbr.4	⊢ ((ψ ∧ θ ∧ η) → ζ)

Assertion

Ref	Expression
syl3anbr	⊢ ((φ ∧ χ ∧ τ) → ζ)

Proof of Theorem syl3anbr

Step	Hyp	Ref	Expression
1		syl3anbr.1	. . 3 ⊢ (ψ ↔ φ)
2	1	bicomi 193	. 2 ⊢ (φ ↔ ψ)
3		syl3anbr.2	. . 3 ⊢ (θ ↔ χ)
4	3	bicomi 193	. 2 ⊢ (χ ↔ θ)
5		syl3anbr.3	. . 3 ⊢ (η ↔ τ)
6	5	bicomi 193	. 2 ⊢ (τ ↔ η)
7		syl3anbr.4	. 2 ⊢ ((ψ ∧ θ ∧ η) → ζ)
8	2, 4, 6, 7	syl3anb 1225	1 ⊢ ((φ ∧ χ ∧ τ) → ζ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by: (None)