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Metamath Proof Explorer

Theorem bj-19.9htbi

Description: Strengthening 19.9ht by replacing its consequent with a biconditional ( 19.9t does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019)

		Ref	Expression
	Assertion	bj-19.9htbi	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.9ht	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝜑 → 𝜑 ) )
2		19.8a	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )
3	1 2	impbid1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )