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Theorem sb5 2269
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 2088 and even needs no disjoint variable condition, see sb1 2497. Theorem sb5f 2532 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2270. (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 2266 . . 3 𝑥[𝑦 / 𝑥]𝜑
2 sbequ12 2245 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2equsexv 2261 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ [𝑦 / 𝑥]𝜑)
43bicomi 226 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wex 1773  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by:  sb56  2270  2sb5  2275  sbnvOLD  2315  sb7f  2562  sbc2or  3779  sbc5  3798
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