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Metamath Proof Explorer

Theorem 19.9d

Description: A deduction version of one direction of 19.9 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 24-Sep-2016) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by Wolf Lammen, 8-Jul-2022)

		Ref	Expression
	Hypothesis	19.9d.1	⊢ ( 𝜓 → Ⅎ 𝑥 𝜑 )
	Assertion	19.9d	⊢ ( 𝜓 → ( ∃ 𝑥 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.9d.1	⊢ ( 𝜓 → Ⅎ 𝑥 𝜑 )
2	1	nfrd	⊢ ( 𝜓 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
3		sp	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )
4	2 3	syl6	⊢ ( 𝜓 → ( ∃ 𝑥 𝜑 → 𝜑 ) )