Theorem sb8mo 2223

Description: Variable substitution in uniqueness quantifier. (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 2093	. . 3 ⊢ (∃xφ ↔ ∃y[y / x]φ)
3	1	sb8eu 2222	. . 3 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
4	2, 3	imbi12i 316	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ (∃y[y / x]φ → ∃!y[y / x]φ))
5		df-mo 2209	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
6		df-mo 2209	. 2 ⊢ (∃*y[y / x]φ ↔ (∃y[y / x]φ → ∃!y[y / x]φ))
7	4, 5, 6	3bitr4i 268	1 ⊢ (∃*xφ ↔ ∃*y[y / x]φ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)