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Metamath Proof Explorer

Theorem eexinst01

Description: exinst01 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eexinst01.1	⊢ ∃ 𝑥 𝜓
	eexinst01.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	eexinst01.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	eexinst01.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	eexinst01	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		eexinst01.1	⊢ ∃ 𝑥 𝜓
2		eexinst01.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3		eexinst01.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4		eexinst01.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
5	3 4 2	exlimdh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )
6	1 5	mpi	⊢ ( 𝜑 → 𝜒 )