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Theorem pm11.58 37415
Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.58 (∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm11.58
StepHypRef Expression
1 19.8a 2037 . . . . 5 (𝜑 → ∃𝑥𝜑)
2 nfv 1829 . . . . . 6 𝑦𝜑
32sb8e 2412 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
41, 3sylib 206 . . . 4 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
54pm4.71i 661 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6 19.42v 1904 . . 3 (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
75, 6bitr4i 265 . 2 (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
87exbii 1763 1 (∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wex 1694  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by: (None)
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