Theorem sbv 2128

Description: Substitution for a variable not occurring in a proposition. See sbf 2312 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2128 can be proved directly by chaining equsv 2030 with sb6 2125. (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 2122	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1939	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 18	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1937	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2105	. . 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 1565  ∃wex 1806  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098

This theorem is referenced by:  sbcom4  2129  sbrimvw  2131  sbievw  2134  sbievw2  2139  sbabel  2963  sbcg  3825  ab0w  4342  iuninc  32846  measiuns  34552  ballotlemodife  34833  xpab  36151  subsym1  36861  bj-vn0ALT  37630  mptsnunlem  37906  ichv  48121  ichf  48122