Theorem mpd3an3 1274

Description: An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)

Hypotheses

Ref	Expression
mpd3an3.2	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
mpd3an3.3	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
mpd3an3	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem mpd3an3

Step	Hyp	Ref	Expression
1		mpd3an3.2	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2		mpd3an3.3	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	2	3expa 1143	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	1, 3	mpdan 412	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 926

This theorem is referenced by:  stoic2b  1364  elovmpt2  5845  oav  6215  omv  6216  oeiv  6217  f1oeng  6474  mulpipq2  6930  ltrnqg  6979  genipv  7068  subval  7674  subap0  8118  fzrevral3  9521  fzoval  9559  subsq2  10062  bcval  10157  dvdsmul1  11096  dvdsmul2  11097  gcdval  11229  eucalgval2  11313  setsvalg  11524  cncfval  11628