Theorem pm5.32da 622

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)

Hypothesis

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

Assertion

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

Proof of Theorem pm5.32da

Step	Hyp	Ref	Expression
1		pm5.32da.1	. . 3 ⊢ ((φ ∧ ψ) → (χ ↔ θ))
2	1	ex 423	. 2 ⊢ (φ → (ψ → (χ ↔ θ)))
3	2	pm5.32d 620	1 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  rexbida  2630  reubida  2794  rmobida  2799  fcnvres  5244  funbrfv2b  5363  dffn5  5364  fnrnfv  5365  fniniseg  5372  eqfnfv2  5394  funiunfv  5468  dff13  5472  mpteq12f  5656  mpt2eq3dva  5670  ltlenlec  6208