Theorem wl-sb8motv 37786

Description: Substitution of variable in universal quantifier. Closed form of sb8mo 2601 without ax-13 2376, but requiring 𝑥 and 𝑦 being disjoint.

This theorem relates to wl-mo3t 37781, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2276 and sbco 2511. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 37781 in a simple fashion. From an educational standpoint, one would assume wl-mo3t 37781 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 3-May-2025.)

Assertion

Ref	Expression
wl-sb8motv	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-sb8motv

Step	Hyp	Ref	Expression
1		wl-sb8eft 37756	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
2		wl-sb8eutv 37784	. . 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 1539  ∃wex 1780  Ⅎwnf 1784  [wsb 2067  ∃*wmo 2537  ∃!weu 2568

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 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569

This theorem is referenced by: (None)