Theorem mpan22 532

Description: An inference based on modus ponens.

Hypotheses

Ref	Expression
mpan22.1	⊢ χ
mpan22.2	⊢ ((φ ∧ (ψ ∧ χ)) → θ)

Assertion

Ref	Expression
mpan22	⊢ ((φ ∧ ψ) → θ)

Proof of Theorem mpan22

Step	Hyp	Ref	Expression
1		mpan22.1	. . 3 ⊢ χ
2		mpan22.2	. . . 4 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
3	2	exp 291	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
4	1, 3	mpan2i 522	. 2 ⊢ (φ → (ψ → θ))
5	4	imp 277	1 ⊢ ((φ ∧ ψ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  aceq6b 3565  prlem934b 3932  rimul 4534

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  df-an 198

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