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Theorem sbidm 1776
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
StepHypRef Expression
1 df-sb 1690 . . . . 5 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simplbi 268 . . . 4 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
32sbimi 1691 . . 3 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
4 sbequ8 1772 . . 3 ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
53, 4sylibr 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 1693 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
8 pm4.24 387 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
9 ax-ie1 1425 . . . . . . . . 9 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
10919.41h 1618 . . . . . . . 8 (∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
118, 10bitr4i 185 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
12 ax-1 5 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑦𝜑))
1312anim2i 334 . . . . . . . . 9 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)))
1413anim1i 333 . . . . . . . 8 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
1514eximi 1534 . . . . . . 7 (∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
1611, 15sylbi 119 . . . . . 6 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
17 anass 393 . . . . . . 7 (((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
1817exbii 1539 . . . . . 6 (∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
1916, 18sylib 120 . . . . 5 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
201anbi2i 445 . . . . . 6 ((𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
2120exbii 1539 . . . . 5 (∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
2219, 21sylibr 132 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
237, 22syl 14 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
24 df-sb 1690 . . 3 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)))
256, 23, 24sylanbrc 408 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
265, 25impbii 124 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1424  [wsb 1689
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-sb 1690
This theorem is referenced by: (None)
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