Theorem wl-sb8mot 37603

Description: Substitution of variable in universal quantifier. Closed form of sb8mo 2601. (Contributed by Wolf Lammen, 11-Aug-2019.)

Assertion

Ref	Expression
wl-sb8mot	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8mot

Step	Hyp	Ref	Expression
1		wl-sb8et 37576	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
2		wl-sb8eut 37601	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
3	1, 2	imbi12d 344	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)))
4		moeu 2583	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
5		moeu 2583	. 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
6	3, 4, 5	3bitr4g 314	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783  [wsb 2065  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)