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 2070. (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		df-sb 2070	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		alequexv 2007	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		exsbim 2008	. . 3 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
4	2, 3	syl 17	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑥𝜑)
5	1, 4	sylbi 219	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-sb 2070

This theorem is referenced by:  sbv  2098  sbft  2270  sb1  2503  2mo  2733  noel  4296  bj-sbft  34104  wl-lem-moexsb  34819  nsb  39121  spsbce-2  40733  sb5ALT  40879  sb5ALTVD  41267