Theorem spsbeOLD 2089

Description: Obsolete version of spsbe 2088 as of 11-Jul-2023. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem spsbeOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sb 2070	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		ax6ev 1972	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑡
3		exim 1834	. . . 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 219	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
6		exim 1834	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
7	6	eximi 1835	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
8		ax6ev 1972	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
9		pm2.27 42	. . . 4 ⊢ (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑))
10	8, 9	ax-mp 5	. . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
11	10	exlimiv 1931	. 2 ⊢ (∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
12	5, 7, 11	3syl 18	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: (None)