Theorem com45 97

Description: Commutation of antecedents. Swap 4th and 5th. Deduction associated with com34 91. Double deduction associated with com23 86. Triple deduction associated with com12 32. (Contributed by Jeff Hankins, 28-Jun-2009.)

Hypothesis

Ref	Expression
com5.1	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))))

Assertion

Ref	Expression
com45	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃 → 𝜂)))))

Proof of Theorem com45

Step	Hyp	Ref	Expression
1		com5.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))))
2		pm2.04 90	. 2 ⊢ ((𝜃 → (𝜏 → 𝜂)) → (𝜏 → (𝜃 → 𝜂)))
3	1, 2	syl8 76	1 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃 → 𝜂)))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  com35  98  com25  99  com5l  100  lcmfdvdsb  16680  prmgaplem6  17094  islmhm2  21037  dfon2lem8  35791