Theorem bisyl7 189

Description: A mixed syllogism inference from a doubly nested implication and a biconditional.

Hypotheses

Ref	Expression
bisyl7.1	⊢ (φ → (ψ → (χ → θ)))
bisyl7.2	⊢ (τ ↔ χ)

Assertion

Ref	Expression
bisyl7	⊢ (φ → (ψ → (τ → θ)))

Proof of Theorem bisyl7

Step	Hyp	Ref	Expression
1		bisyl7.1	. 2 ⊢ (φ → (ψ → (χ → θ)))
2		bisyl7.2	. . 3 ⊢ (τ ↔ χ)
3	2	biimp 133	. 2 ⊢ (τ → χ)
4	1, 3	syl7 24	1 ⊢ (φ → (ψ → (τ → θ)))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  jao 274  zfpair 1891

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

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