Theorem bj-ssbft 32642

Description: See sbft 2379. This proof is from Tarski's FOL together with sp 2053 (and its dual). (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem bj-ssbft

Step	Hyp	Ref	Expression
1		bj-sbex 32626	. . 3 ⊢ ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
2		df-nf 1710	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	2	biimpi 206	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4		sp 2053	. . 3 ⊢ (∀𝑥𝜑 → 𝜑)
5	1, 3, 4	syl56 36	. 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑 → 𝜑))
6		19.8a 2052	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
7		bj-alsb 32625	. . 3 ⊢ (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
8	6, 3, 7	syl56 36	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑡/𝑥]b𝜑))
9	5, 8	impbid 202	1 ⊢ (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-ssb 32620

This theorem is referenced by: (None)