Theorem pm5.74da 668

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)

Hypothesis

Ref	Expression
pm5.74da.1	⊢ ((φ ∧ ψ) → (χ ↔ θ))

Assertion

Ref	Expression
pm5.74da	⊢ (φ → ((ψ → χ) ↔ (ψ → θ)))

Proof of Theorem pm5.74da

Step	Hyp	Ref	Expression
1		pm5.74da.1	. . 3 ⊢ ((φ ∧ ψ) → (χ ↔ θ))
2	1	ex 423	. 2 ⊢ (φ → (ψ → (χ ↔ θ)))
3	2	pm5.74d 238	1 ⊢ (φ → ((ψ → χ) ↔ (ψ → θ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  ralbida  2629