Theorem sbv 2069

Description: Substitution for a variable not occurring in a proposition. See sbf 2234 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2069 can be proved directly by chaining equsv 1986 with sb6 2066. (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 2061	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1890	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 17	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1888	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2046	. . 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 1520  ∃wex 1761  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947

This theorem depends on definitions:  df-bi 208  df-ex 1762  df-sb 2043

This theorem is referenced by:  sbcom4  2070  sbievw2  2074  iuninc  30002  measiuns  31093  ballotlemodife  31372  bj-vjust  33944  mptsnunlem  34150  ichv  43091  ichf  43092