Theorem sbv 2099

Description: Substitution for a variable not occurring in a proposition. See sbf 2282 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2099 can be proved directly by chaining equsv 2010 with sb6 2096. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
sbv	⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑡)

Proof of Theorem sbv

Step	Hyp	Ref	Expression
1		spsbe 2093	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1919	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 17	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1917	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2079	. . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → [𝑡 / 𝑥]𝜑)
7	3, 6	impbii 210	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207  ∀wal 1545  ∃wex 1786  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074

This theorem is referenced by:  sbcom4  2100  sbievw  2104  sbievw2  2109  sbabel  2933  sbcg  3795  ab0w  4307  iuninc  32649  measiuns  34401  ballotlemodife  34682  xpab  35954  subsym1  36655  mptsnunlem  37700  ichv  47924  ichf  47925