Theorem syl3an1br 625

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	⊢ ((φ ∧ ψ ∧ χ) → θ)
syl3an1br.2	⊢ (φ ↔ τ)

Assertion

Ref	Expression
syl3an1br	⊢ ((τ ∧ ψ ∧ χ) → θ)

Proof of Theorem syl3an1br

Step	Hyp	Ref	Expression
1		syl3an.1	. 2 ⊢ ((φ ∧ ψ ∧ χ) → θ)
2		syl3an1br.2	. . 3 ⊢ (φ ↔ τ)
3	2	biimpr 134	. 2 ⊢ (τ → φ)
4	1, 3	syl3an1 619	1 ⊢ ((τ ∧ ψ ∧ χ) → θ)

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

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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