Theorem mpd3an23 1276

Description: An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)

Hypotheses

Ref	Expression
mpd3an23.1	⊢ (𝜑 → 𝜓)
mpd3an23.2	⊢ (𝜑 → 𝜒)
mpd3an23.3	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
mpd3an23	⊢ (𝜑 → 𝜃)

Proof of Theorem mpd3an23

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (𝜑 → 𝜑)
2		mpd3an23.1	. 2 ⊢ (𝜑 → 𝜓)
3		mpd3an23.2	. 2 ⊢ (𝜑 → 𝜒)
4		mpd3an23.3	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
5	1, 2, 3, 4	syl3anc 1175	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ w3a 925

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 927

This theorem is referenced by:  exp0  10013  bcpasc  10228  bccl  10229  pw2dvds  11476  qnumdencoprm  11503  qeqnumdivden  11504