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Theorem moanim 1971
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
Hypothesis
Ref Expression
moanim.1 xφ
Assertion
Ref Expression
moanim (∃*x(φ ψ) ↔ (φ∃*xψ))

Proof of Theorem moanim
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 anandi 524 . . . . 5 ((φ (ψ [y / x]ψ)) ↔ ((φ ψ) (φ [y / x]ψ)))
21imbi1i 227 . . . 4 (((φ (ψ [y / x]ψ)) → x = y) ↔ (((φ ψ) (φ [y / x]ψ)) → x = y))
3 impexp 250 . . . 4 (((φ (ψ [y / x]ψ)) → x = y) ↔ (φ → ((ψ [y / x]ψ) → x = y)))
4 sban 1826 . . . . . . 7 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
5 moanim.1 . . . . . . . . 9 xφ
65sbf 1657 . . . . . . . 8 ([y / x]φφ)
76anbi1i 431 . . . . . . 7 (([y / x]φ [y / x]ψ) ↔ (φ [y / x]ψ))
84, 7bitr2i 174 . . . . . 6 ((φ [y / x]ψ) ↔ [y / x](φ ψ))
98anbi2i 430 . . . . 5 (((φ ψ) (φ [y / x]ψ)) ↔ ((φ ψ) [y / x](φ ψ)))
109imbi1i 227 . . . 4 ((((φ ψ) (φ [y / x]ψ)) → x = y) ↔ (((φ ψ) [y / x](φ ψ)) → x = y))
112, 3, 103bitr3i 199 . . 3 ((φ → ((ψ [y / x]ψ) → x = y)) ↔ (((φ ψ) [y / x](φ ψ)) → x = y))
12112albii 1357 . 2 (xy(φ → ((ψ [y / x]ψ) → x = y)) ↔ xy(((φ ψ) [y / x](φ ψ)) → x = y))
13519.21 1472 . . 3 (x(φy((ψ [y / x]ψ) → x = y)) ↔ (φxy((ψ [y / x]ψ) → x = y)))
14 19.21v 1750 . . . 4 (y(φ → ((ψ [y / x]ψ) → x = y)) ↔ (φy((ψ [y / x]ψ) → x = y)))
1514albii 1356 . . 3 (xy(φ → ((ψ [y / x]ψ) → x = y)) ↔ x(φy((ψ [y / x]ψ) → x = y)))
16 ax-17 1416 . . . . 5 (ψyψ)
1716mo3h 1950 . . . 4 (∃*xψxy((ψ [y / x]ψ) → x = y))
1817imbi2i 215 . . 3 ((φ∃*xψ) ↔ (φxy((ψ [y / x]ψ) → x = y)))
1913, 15, 183bitr4ri 202 . 2 ((φ∃*xψ) ↔ xy(φ → ((ψ [y / x]ψ) → x = y)))
20 ax-17 1416 . . 3 ((φ ψ) → y(φ ψ))
2120mo3h 1950 . 2 (∃*x(φ ψ) ↔ xy(((φ ψ) [y / x](φ ψ)) → x = y))
2212, 19, 213bitr4ri 202 1 (∃*x(φ ψ) ↔ (φ∃*xψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346  [wsb 1642  ∃*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-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  moanimv  1972  moaneu  1973  moanmo  1974
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