Theorem syld3an2 1318

Description: A syllogism inference. (Contributed by NM, 20-May-2007.)

Hypotheses

Ref	Expression
syld3an2.1	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓)
syld3an2.2	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syld3an2	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Proof of Theorem syld3an2

Step	Hyp	Ref	Expression
1		syld3an2.1	. . . 4 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜓)
2	1	3com23 1233	. . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜓)
3		syld3an2.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
4	3	3com23 1233	. . 3 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜏)
5	2, 4	syld3an3 1316	. 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → 𝜏)
6	5	3com23 1233	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 1002

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 1004

This theorem is referenced by:  nppcan2  8400  nnncan  8404  nnncan2  8406  ltdivmul  9046  ledivmul  9047  ltdiv23  9062  lediv23  9063  pfxtrcfv  11264  dvdssub2  12386  dvdsgcdb  12574  lcmdvdsb  12646  ressabsg  13149  mulginvcom  13724  lspssp  14407  rpdivcxp  15625