Theorem mpan21 531

Description: An inference based on modus ponens.

Hypotheses

Ref	Expression
mpan21.1	⊢ ψ
mpan21.2	⊢ ((φ ∧ (ψ ∧ χ)) → θ)

Assertion

Ref	Expression
mpan21	⊢ ((φ ∧ χ) → θ)

Proof of Theorem mpan21

Step	Hyp	Ref	Expression
1		mpan21.1	. . 3 ⊢ ψ
2		mpan21.2	. . . 4 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
3	2	exp 291	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
4	1, 3	mpani 521	. 2 ⊢ (φ → (χ → θ))
5	4	imp 277	1 ⊢ ((φ ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  mpan121 533  tz7.7 2224  tfr3 2964  oacl 3138  omcl 3139  oaordi 3148  oawordri 3152  oaass 3163  omordi 3164  zornlem1 3603  htalem 3618  axcnre 4087

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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