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Theorem sb5 2267
Description: Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2086 and even needs no disjoint variable condition, see sb1 2496. Theorem sb5f 2531 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2268. (Revised by Wolf Lammen, 4-Sep-2023.)
Assertion
Ref Expression
sb5 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5
StepHypRef Expression
1 nfs1v 2264 . . 3 𝑥[𝑦 / 𝑥]𝜑
2 sbequ12 2243 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2equsexv 2259 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ [𝑦 / 𝑥]𝜑)
43bicomi 225 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  sb56  2268  2sb5  2273  sbnvOLD  2313  sb7f  2561  sbc2or  3778  sbc5  3797
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