Theorem mpan11 529

Description: An inference based on modus ponens.

Hypotheses

Ref	Expression
mpan11.1	⊢ φ
mpan11.2	⊢ (((φ ∧ ψ) ∧ χ) → θ)

Assertion

Ref	Expression
mpan11	⊢ ((ψ ∧ χ) → θ)

Proof of Theorem mpan11

Step	Hyp	Ref	Expression
1		mpan11.1	. . 3 ⊢ φ
2		mpan11.2	. . . 4 ⊢ (((φ ∧ ψ) ∧ χ) → θ)
3	2	exp 291	. . 3 ⊢ ((φ ∧ ψ) → (χ → θ))
4	1, 3	mpan 518	. 2 ⊢ (ψ → (χ → θ))
5	4	imp 277	1 ⊢ ((ψ ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  tfrlem9 2957  suppsr3 4018  divasst 4239  divnegt 4259  recdivt 4270  divdiv23t 4271  rimul 4534  h1datom 5483  strlem1 5691

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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