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Metamath Proof Explorer

Theorem eexinst11

Description: exinst11 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eexinst11.1	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
	eexinst11.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	eexinst11.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	eexinst11.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	eexinst11	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		eexinst11.1	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
2		eexinst11.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3		eexinst11.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4		eexinst11.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
5	3 4 2	exlimdh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )
6	1 5	syl5com	⊢ ( 𝜑 → ( 𝜑 → 𝜒 ) )
7	6	pm2.43i	⊢ ( 𝜑 → 𝜒 )