Theorem sbv 2121

Description: Substitution for a variable not occurring in a proposition. See sbf 2305 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2121 can be proved directly by chaining equsv 2023 with sb6 2118. (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 2115	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2		ax5e 1932	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
3	1, 2	syl 17	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → 𝜑)
4		ax-5 1930	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		stdpc4 2098	. . 3 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → [𝑡 / 𝑥]𝜑)
7	3, 6	impbii 211	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1558  ∃wex 1799  [wsb 2090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091

This theorem is referenced by:  sbcom4  2122  sbrimvw  2124  sbievw  2127  sbievw2  2132  sbabel  2956  sbcg  3816  ab0w  4332  iuninc  32760  measiuns  34514  ballotlemodife  34795  xpab  36076  subsym1  36787  mptsnunlem  37832  ichv  48055  ichf  48056