Theorem spsbe 2033

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 2016. (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 2016	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		ax6ev 1930	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑡
3		exim 1796	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦 𝑦 = 𝑡 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	2, 3	mpi 20	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 4	sylbi 209	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
6		exsbim 1958	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
7	5, 6	syl 17	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1505  ∃wex 1742  [wsb 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928

This theorem depends on definitions:  df-bi 199  df-ex 1743  df-sb 2016

This theorem is referenced by:  sbv  2040  sbft  2198  2mo  2680  noel  4184  bj-ssbft  33500  wl-lem-moexsb  34242  nsb  38542  spsbce-2  40126  sb5ALT  40275  sb5ALTVD  40663