pm5.32da - New Foundations Explorer

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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)))

Colors of variables: wff setvar class

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

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

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

This theorem is referenced by: rexbida 2629 reubida 2793 rmobida 2798 fcnvres 5243 funbrfv2b 5362 dffn5 5363 fnrnfv 5364 fniniseg 5371 eqfnfv2 5393 funiunfv 5467 dff13 5471 mpteq12f 5655 mpt2eq3dva 5669 ltlenlec 6207