Theorem spsbe 2115

Description: Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2091. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) Revise df-sb 2091. (Revised by Wolf Lammen, 4-Jun-2026.)

Assertion

Ref	Expression
spsbe	⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbe

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbimp 2092	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		alequexv 2021	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		exsbim 2022	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
4	1, 2, 3	3syl 18	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀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:  sbv  2121  nsb  2140  sbft  2304  sb1  2509  2mo  2675  bj-sbft  37250  wl-lem-moexsb  38068  spsbce-2  44954  sb5ALT  45098  sb5ALTVD  45485