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Theorem wl-dfrexsb 35167
 Description: An alternate definition of restricted existential quantification (df-wl-rex 35162) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
Assertion
Ref Expression
wl-dfrexsb (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfrexsb
StepHypRef Expression
1 df-wl-rex 35162 . . 3 (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
2 wl-dfralsb 35153 . . 3 (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
31, 2xchbinx 337 . 2 (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
4 sbn 2283 . . . . . 6 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
54imbi2i 339 . . . . 5 ((𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦𝐴 → ¬ [𝑦 / 𝑥]𝜑))
65albii 1821 . . . 4 (∀𝑦(𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦(𝑦𝐴 → ¬ [𝑦 / 𝑥]𝜑))
7 imnang 1843 . . . 4 (∀𝑦(𝑦𝐴 → ¬ [𝑦 / 𝑥]𝜑) ↔ ∀𝑦 ¬ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
8 alnex 1783 . . . 4 (∀𝑦 ¬ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ¬ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
96, 7, 83bitri 300 . . 3 (∀𝑦(𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ¬ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
109con2bii 361 . 2 (∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ¬ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥] ¬ 𝜑))
113, 10bitr4i 281 1 (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  [wsb 2069   ∈ wcel 2111  ∀wl-ral 35147  ∃wl-rex 35148 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-wl-ral 35152  df-wl-rex 35162 This theorem is referenced by:  wl-dfreusb  35173
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