Theorem pm5.32d 620

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.)

Hypothesis

Ref	Expression
pm5.32d.1	⊢ (φ → (ψ → (χ ↔ θ)))

Assertion

Ref	Expression
pm5.32d	⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))

Proof of Theorem pm5.32d

Step	Hyp	Ref	Expression
1		pm5.32d.1	. 2 ⊢ (φ → (ψ → (χ ↔ θ)))
2		pm5.32 617	. 2 ⊢ ((ψ → (χ ↔ θ)) ↔ ((ψ ∧ χ) ↔ (ψ ∧ θ)))
3	1, 2	sylib 188	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:  pm5.32rd  621  pm5.32da  622  anbi2d  684  cbval2  2004  cbvex2  2005  cores  5085  isoini  5498  mpt2eq123  5662  nmembers1lem3  6271