Theorem bj-ceqsalg0 34197

Description: The FOL content of ceqsalg 3528. (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 1790	. 2 ⊢ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓))
4		bj-ceqsalt0 34193	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜒) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))
5	1, 3, 4	mp3an12 1445	1 ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1529  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1084  df-ex 1775  df-nf 1779

This theorem is referenced by:  bj-ceqsalg  34198  bj-ceqsalgv  34200