Theorem sbv 2122

Description: Substitution for a variable not occurring in a proposition. See sbf 2306 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2122 can be proved directly by chaining equsv 2033 with sb6 2119. (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 2116	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1942	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 18	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1940	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2102	. . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → [𝑡 / 𝑥]𝜑)
7	3, 6	impbii 212	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1568  ∃wex 1809  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097

This theorem is referenced by:  sbcom4  2123  sbrimvw  2125  sbievw  2128  sbievw2  2133  sbabel  2957  sbcg  3816  ab0w  4335  iuninc  32905  measiuns  34607  ballotlemodife  34888  xpab  36218  subsym1  36958  bj-vn0ALT  37728  mptsnunlem  38004  ichv  48218  ichf  48219