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Metamath Proof Explorer

Theorem com45

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

		Ref	Expression
	Hypothesis	com5.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → ( 𝜏 → 𝜂 ) ) ) ) )
	Assertion	com45	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 → ( 𝜃 → 𝜂 ) ) ) ) )

Proof

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