Theorem sbv 2091

Description: Substitution for a variable not occurring in a proposition. See sbf 2273 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2091 can be proved directly by chaining equsv 2004 with sb6 2088. (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 2085	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1913	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 17	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1911	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2071	. . 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 1539  ∃wex 1780  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-sb 2068

This theorem is referenced by:  sbcom4  2092  sbievw  2096  sbievw2  2101  sbabel  2927  sbcg  3809  ab0w  4326  ralf0  4461  iuninc  32540  measiuns  34230  ballotlemodife  34511  xpab  35770  subsym1  36469  mptsnunlem  37380  ichv  47488  ichf  47489