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Theorem pm11.58 40873
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 2180 . . . . 5 (𝜑 → ∃𝑥𝜑)
2 nfv 1915 . . . . . 6 𝑦𝜑
32sb8e 2560 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
41, 3sylib 220 . . . 4 (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
54pm4.71i 562 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6 19.42v 1954 . . 3 (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
75, 6bitr4i 280 . 2 (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
87exbii 1848 1 (∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wex 1780  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070
This theorem is referenced by: (None)
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