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Theorem wl-sb8mot 37561
Description: Substitution of variable in universal quantifier. Closed form of sb8mo 2599. (Contributed by Wolf Lammen, 11-Aug-2019.)
Assertion
Ref Expression
wl-sb8mot (∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8mot
StepHypRef Expression
1 wl-sb8et 37534 . . 3 (∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
2 wl-sb8eut 37559 . . 3 (∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
31, 2imbi12d 344 . 2 (∀𝑥𝑦𝜑 → ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)))
4 moeu 2581 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
5 moeu 2581 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
63, 4, 53bitr4g 314 1 (∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wex 1776  wnf 1780  [wsb 2062  ∃*wmo 2536  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567
This theorem is referenced by: (None)
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