Theorem sbidm 1772

Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Assertion

Ref	Expression
sbidm	⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbidm

Step	Hyp	Ref	Expression
1		df-sb 1686	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simplbi 268	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → 𝜑))
3	2	sbimi 1687	. . 3 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑))
4		sbequ8 1768	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑))
5	3, 4	sylibr 132	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)
6		ax-1 5	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
7		sb1 1689	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8		pm4.24 387	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
9		ax-ie1 1422	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	9	19.41h 1615	. . . . . . . 8 ⊢ (∃𝑥((𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
11	8, 10	bitr4i 185	. . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥((𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
12		ax-1 5	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
13	12	anim2i 334	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)))
14	13	anim1i 333	. . . . . . . 8 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
15	14	eximi 1531	. . . . . . 7 ⊢ (∃𝑥((𝑥 = 𝑦 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
16	11, 15	sylbi 119	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
17		anass 393	. . . . . . 7 ⊢ (((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))))
18	17	exbii 1536	. . . . . 6 ⊢ (∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))))
19	16, 18	sylib 120	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))))
20	1	anbi2i 444	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))))
21	20	exbii 1536	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))))
22	19, 21	sylibr 132	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
23	7, 22	syl 14	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
24		df-sb 1686	. . 3 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)))
25	6, 23, 24	sylanbrc 408	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
26	5, 25	impbii 124	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∃wex 1421  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)