Theorem syl6com 35

Description: Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Hypotheses

Ref	Expression
syl6com.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl6com.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
syl6com	⊢ (𝜓 → (𝜑 → 𝜃))

Proof of Theorem syl6com

Step	Hyp	Ref	Expression
1		syl6com.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		syl6com.2	. . 3 ⊢ (𝜒 → 𝜃)
3	1, 2	syl6 33	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	com12 30	1 ⊢ (𝜓 → (𝜑 → 𝜃))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  pclem6  1423  spimh  1790  ax16  1866  ax16i  1911  elres  5101  funcnvuni  5453  funrnex  6347  negf1o  8717  lidrididd  13762  dfgrp2  13892  rngdi  14347  rngdir  14348  basis2  15286  clwwlknun  16899  bj-inf2vnlem2  17214