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Theorem pm11.58 38907
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 2090 . . . . 5 (𝜑 → ∃𝑥𝜑)
2 nfv 1883 . . . . . 6 𝑦𝜑
32sb8e 2453 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
41, 3sylib 208 . . . 4 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
54pm4.71i 665 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6 19.42v 1921 . . 3 (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
75, 6bitr4i 267 . 2 (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
87exbii 1814 1 (∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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