Theorem syl-P3.24 259

Description: Syllogism. †

Hypotheses

Ref	Expression
syl-P3.24.1	⊢ (𝛾 → (𝜑 → 𝜓))
syl-P3.24.2	⊢ (𝛾 → (𝜓 → 𝜒))

Assertion

Ref	Expression
syl-P3.24	⊢ (𝛾 → (𝜑 → 𝜒))

Proof of Theorem syl-P3.24

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		syl-P3.24.1	. . . . 5 ⊢ (𝛾 → (𝜑 → 𝜓))
3	2	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → (𝜑 → 𝜓))
4	1, 3	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
5		syl-P3.24.2	. . . 4 ⊢ (𝛾 → (𝜓 → 𝜒))
6	5	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ 𝜑) → (𝜓 → 𝜒))
7	4, 6	ndime-P3.6 171	. 2 ⊢ ((𝛾 ∧ 𝜑) → 𝜒)
8	7	rcp-NDIMI2 224	1 ⊢ (𝛾 → (𝜑 → 𝜒))

Syntax hints:   → wff-imp 10   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133

This theorem is referenced by:  syl-P3.24.RC  260  dsyl-P3.25  261  imsubl-P3.28a  267  imsubr-P3.28b  269  imsubd-P3.28c  271  bitrns-P3.33c  302  tsyl-P4.15  426  qimeqalla-P6-L1  698  qimeqallb-P6-L1  700  exi-P6  718  nfrgencl-L6  811  gennfrcl-L6  812  qremexd-P6  823  exiad-P6  824  axL4-P7  945  axL4ex-P7  946  exia-P7  953  qimeqallb-P7  976  qimeqalla-P7  1050