Theorem sb8mo 1911

Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
sb8eu.1	⊢ Ⅎyφ

Assertion

Ref	Expression
sb8mo	⊢ (∃*xφ ↔ ∃*y[y / x]φ)

Proof of Theorem sb8mo

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . 4 ⊢ Ⅎyφ
2	1	sb8e 1734	. . 3 ⊢ (∃xφ ↔ ∃y[y / x]φ)
3	1	sb8eu 1910	. . 3 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
4	2, 3	imbi12i 228	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ (∃y[y / x]φ → ∃!y[y / x]φ))
5		df-mo 1901	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
6		df-mo 1901	. 2 ⊢ (∃*y[y / x]φ ↔ (∃y[y / x]φ → ∃!y[y / x]φ))
7	4, 5, 6	3bitr4i 201	1 ⊢ (∃*xφ ↔ ∃*y[y / x]φ)

Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1346  ∃wex 1378  [wsb 1642  ∃!weu 1897  ∃*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by: (None)