Theorem mpd3an23 1349

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 1248	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ w3a 979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 981

This theorem is referenced by:  exp0  10538  bcpasc  10760  bccl  10761  pw2dvds  12180  qnumdencoprm  12207  qeqnumdivden  12208  grpinvid  12965  ghmid  13144  mgpvalg  13232  mgpex  13234  opprex  13378  unitgrpid  13423  qusmul2  13773  dvef  14541