Theorem bj-ceqsalg0 35000

Description: The FOL content of ceqsalg 3454. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsalg0.1	⊢ Ⅎ𝑥𝜓
bj-ceqsalg0.2	⊢ (𝜒 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-ceqsalg0	⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Proof of Theorem bj-ceqsalg0

Step	Hyp	Ref	Expression
1		bj-ceqsalg0.1	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-ceqsalg0.2	. . 3 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
3	2	ax-gen 1799	. 2 ⊢ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓))
4		bj-ceqsalt0 34996	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜒) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
5	1, 3, 4	mp3an12 1449	1 ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ex 1784  df-nf 1788

This theorem is referenced by:  bj-ceqsalg  35001  bj-ceqsalgv  35003