Theorem sbv 2121

Description: Substitution for a variable not occurring in a proposition. See sbf 2305 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2121 can be proved directly by chaining equsv 2032 with sb6 2118. (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 2115	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1941	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 18	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1939	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2101	. . 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)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  ∃wex 1808  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096

This theorem is used by:  sbcom4  2122  sbrimvw  2124  sbievw  2127  sbievw2  2132  sbabel  2956  sbcg  3815  ab0w  4334  iuninc  32916  measiuns  34616  ballotlemodife  34897  xpab  36226  subsym1  36966  bj-vn0ALT  37736  mptsnunlem  38012  ichv  48226  ichf  48227