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Theorem wl-dv-sb 33820
Description: Substitution for 𝑥 has no effect on 𝜑 not containing 𝑥. See also sbf 2497. (Contributed by Wolf Lammen, 4-Sep-2022.)
Assertion
Ref Expression
wl-dv-sb (𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wl-dv-sb
StepHypRef Expression
1 ax-1 6 . . . 4 (𝜑 → (𝑥 = 𝑦𝜑))
2 ax6e 2390 . . . . . 6 𝑥 𝑥 = 𝑦
32jctl 520 . . . . 5 (𝜑 → (∃𝑥 𝑥 = 𝑦𝜑))
4 19.41v 2045 . . . . 5 (∃𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥 𝑥 = 𝑦𝜑))
53, 4sylibr 226 . . . 4 (𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
61, 5jca 508 . . 3 (𝜑 → ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
7 df-sb 2065 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
86, 7sylibr 226 . 2 (𝜑 → [𝑦 / 𝑥]𝜑)
9 spsbe 2068 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
10 ax5e 2008 . . 3 (∃𝑥𝜑𝜑)
119, 10syl 17 . 2 ([𝑦 / 𝑥]𝜑𝜑)
128, 11impbii 201 1 (𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wex 1875  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-sb 2065
This theorem is referenced by: (None)
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