Theorem bj-ssbft 31831

Description: See sbft 2367. This proof is from Tarski's FOL together with sp 2041 (and its dual). (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbft	⊢ (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑 ↔ 𝜑))

Proof of Theorem bj-ssbft

Step	Hyp	Ref	Expression
1		bj-sbex 31815	. . 3 ⊢ ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
2		df-bj-nf 31765	. . . 4 ⊢ (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	2	biimpi 205	. . 3 ⊢ (ℲℲ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		sp 2041	. . 3 ⊢ (∀𝑥𝜑 → 𝜑)
5	1, 3, 4	syl56 35	. 2 ⊢ (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑 → 𝜑))
6		19.8a 2039	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
7		bj-alsb 31814	. . 3 ⊢ (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
8	6, 3, 7	syl56 35	. 2 ⊢ (ℲℲ𝑥𝜑 → (𝜑 → [𝑡/𝑥]b𝜑))
9	5, 8	impbid 201	1 ⊢ (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  ℲℲwnff 31764  [wssb 31808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-bj-nf 31765  df-ssb 31809

This theorem is referenced by: (None)