Theorem spsbe 2118

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 2099. (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 2100	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		alequexv 2033	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	sylbi 220	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
4		exsbim 2034	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
5	3, 4	syl 18	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  [wsb 2098

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 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099

This theorem is used by:  sbv  2124  nsb  2142  sbft  2303  sb1  2507  2mo  2673  bj-sbft  37776  wl-lem-moexsb  38609  spsbce-2  45474  sb5ALT  45616  sb5ALTVD  46003