Theorem syl3an3b 624

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	⊢ ((φ ∧ ψ ∧ χ) → θ)
syl3an3b.2	⊢ (τ ↔ χ)

Assertion

Ref	Expression
syl3an3b	⊢ ((φ ∧ ψ ∧ τ) → θ)

Proof of Theorem syl3an3b

Step	Hyp	Ref	Expression
1		syl3an.1	. 2 ⊢ ((φ ∧ ψ ∧ χ) → θ)
2		syl3an3b.2	. . 3 ⊢ (τ ↔ χ)
3	2	biimp 133	. 2 ⊢ (τ → χ)
4	1, 3	syl3an3 621	1 ⊢ ((φ ∧ ψ ∧ τ) → θ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ w3a 581

This theorem is referenced by:  nnmsucr 3182

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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