Theorem dfsb1 2484

Description: Alternate definition of substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). This was the original definition before df-sb 2073. Note that it does not require dummy variables in its definiens; this is done by having 𝑥 free in the first conjunct and bound in the second. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by BJ, 9-Jul-2023.) Revise df-sb 2073. (Revised by Wolf Lammen, 29-Jul-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
dfsb1	⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Proof of Theorem dfsb1

Step	Hyp	Ref	Expression
1		sbequ2 2248	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
2	1	com12 32	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
3		sb1 2478	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
4	2, 3	jca 515	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
6		sbequ1 2247	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
7	5, 6	embantd 59	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
8	7	sps 2184	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
9	8	adantrd 495	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑))
10		sb3 2477	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
11	10	adantld 494	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑))
12	9, 11	pm2.61i 185	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → [𝑦 / 𝑥]𝜑)
13	4, 12	impbii 212	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787  [wsb 2072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-12 2177  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2073

This theorem is referenced by:  drsb1  2498  subsym1  34302  bj-dfsb2  34707  frege55b  41123