Theorem sbv 2125

Description: Substitution for a variable not occurring in a proposition. See sbf 2304 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2125 can be proved directly by chaining equsv 2036 with sb6 2122. (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 2119	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1945	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 18	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1943	. . 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)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  sbcom4  2126  sbrimvw  2128  sbievw  2131  sbievw2  2135  sbabel  2954  sbcg  3811  ab0w  4328  iuninc  33037  measiuns  34731  ballotlemodife  35012  xpab  36308  subsym1  37049  bj-vn0ALT  37819  mptsnunlem  38095  ichv  48352  ichf  48353