Theorem sbmo 2234

Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
sbmo	⊢ ([y / x]∃*zφ ↔ ∃*z[y / x]φ)

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbmo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbex 2128	. . 3 ⊢ ([y / x]∃w∀z(φ → z = w) ↔ ∃w[y / x]∀z(φ → z = w))
2		nfv 1619	. . . . . 6 ⊢ Ⅎx z = w
3	2	sblim 2068	. . . . 5 ⊢ ([y / x](φ → z = w) ↔ ([y / x]φ → z = w))
4	3	sbalv 2129	. . . 4 ⊢ ([y / x]∀z(φ → z = w) ↔ ∀z([y / x]φ → z = w))
5	4	exbii 1582	. . 3 ⊢ (∃w[y / x]∀z(φ → z = w) ↔ ∃w∀z([y / x]φ → z = w))
6	1, 5	bitri 240	. 2 ⊢ ([y / x]∃w∀z(φ → z = w) ↔ ∃w∀z([y / x]φ → z = w))
7		nfv 1619	. . . 4 ⊢ Ⅎwφ
8	7	mo2 2233	. . 3 ⊢ (∃*zφ ↔ ∃w∀z(φ → z = w))
9	8	sbbii 1653	. 2 ⊢ ([y / x]∃*zφ ↔ [y / x]∃w∀z(φ → z = w))
10		nfv 1619	. . 3 ⊢ Ⅎw[y / x]φ
11	10	mo2 2233	. 2 ⊢ (∃*z[y / x]φ ↔ ∃w∀z([y / x]φ → z = w))
12	6, 9, 11	3bitr4i 268	1 ⊢ ([y / x]∃*zφ ↔ ∃*z[y / x]φ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃*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)