Theorem spsbe 2088

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 2069. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.)

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		dfsb 2070	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		alequexv 2003	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	sylbi 217	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
4		exsbim 2004	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
5	3, 4	syl 17	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069

This theorem is referenced by:  sbv  2094  nsb  2112  sbft  2277  sb1  2483  2mo  2649  bj-sbft  37018  wl-lem-moexsb  37823  spsbce-2  44737  sb5ALT  44881  sb5ALTVD  45268