Theorem sbv 2089

Description: Substitution for a variable not occurring in a proposition. See sbf 2271 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2089 can be proved directly by chaining equsv 2003 with sb6 2086. (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 2083	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1912	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 17	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1910	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2069	. . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → [𝑡 / 𝑥]𝜑)
7	3, 6	impbii 209	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538  ∃wex 1779  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-sb 2066

This theorem is referenced by:  sbcom4  2090  sbievw  2094  sbievw2  2099  sbabel  2924  sbcg  3823  vn0  4304  eq0  4309  ab0w  4338  ab0orv  4342  eq0rdv  4366  rzal  4468  ralf0  4473  iuninc  32462  measiuns  34180  ballotlemodife  34462  xpab  35686  subsym1  36388  mptsnunlem  37299  ichv  47423  ichf  47424