Theorem sbv 2126

Description: Substitution for a variable not occurring in a proposition. See sbf 2307 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2126 can be proved directly by chaining equsv 2037 with sb6 2123. (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 2120	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1946	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 18	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1944	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2106	. . 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 1811  [wsb 2100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101

This theorem is used by:  sbcom4  2127  sbrimvw  2129  sbievw  2132  sbievw2  2136  sbabel  2958  sbcg  3814  noel  4286  ab0w  4331  iuninc  33212  measiuns  34906  ballotlemodife  35186  xpab  36621  subsym1  37345  bj-vn0ALT  38129  mptsnunlem  38404  ichv  48673  ichf  48674