Theorem bj-ceqsalg0

Description: The FOL content of ceqsalg . (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsalg0.1	⊢ Ⅎ 𝑥 𝜓
	bj-ceqsalg0.2	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-ceqsalg0	⊢ ( ∃ 𝑥 𝜒 → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalg0.1	⊢ Ⅎ 𝑥 𝜓
2		bj-ceqsalg0.2	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3	2	ax-gen	⊢ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
4		bj-ceqsalt0	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜒 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) ↔ 𝜓 ) )
5	1 3 4	mp3an12	⊢ ( ∃ 𝑥 𝜒 → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) ↔ 𝜓 ) )