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Theorem sbidm 1731
 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 1646 . . . . 5 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simplbi 259 . . . 4 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
32sbimi 1647 . . 3 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
4 sbequ8 1727 . . 3 ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
53, 4sylibr 137 . 2 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜑)
6 ax-1 5 . . 3 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
7 sb1 1649 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
8 pm4.24 375 . . . . . . . 8 (∃𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
9 ax-ie1 1382 . . . . . . . . 9 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
10919.41h 1575 . . . . . . . 8 (∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (∃𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
118, 10bitr4i 176 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
12 ax-1 5 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑦𝜑))
1312anim2i 324 . . . . . . . . 9 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)))
1413anim1i 323 . . . . . . . 8 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
1514eximi 1491 . . . . . . 7 (∃𝑥((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
1611, 15sylbi 114 . . . . . 6 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
17 anass 381 . . . . . . 7 (((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
1817exbii 1496 . . . . . 6 (∃𝑥((𝑥 = 𝑦 ∧ (𝑥 = 𝑦𝜑)) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
1916, 18sylib 127 . . . . 5 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
201anbi2i 430 . . . . . 6 ((𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
2120exbii 1496 . . . . 5 (∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑))))
2219, 21sylibr 137 . . . 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 1646 . . 3 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)))
256, 23, 24sylanbrc 394 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
265, 25impbii 117 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1381  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by: (None)
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