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Theorem mo4f 1957
 Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
Hypotheses
Ref Expression
mo4f.1 xψ
mo4f.2 (x = y → (φψ))
Assertion
Ref Expression
mo4f (∃*xφxy((φ ψ) → x = y))
Distinct variable groups:   x,y   φ,y
Allowed substitution hints:   φ(x)   ψ(x,y)

Proof of Theorem mo4f
StepHypRef Expression
1 ax-17 1416 . . 3 (φyφ)
21mo3h 1950 . 2 (∃*xφxy((φ [y / x]φ) → x = y))
3 mo4f.1 . . . . . 6 xψ
4 mo4f.2 . . . . . 6 (x = y → (φψ))
53, 4sbie 1671 . . . . 5 ([y / x]φψ)
65anbi2i 430 . . . 4 ((φ [y / x]φ) ↔ (φ ψ))
76imbi1i 227 . . 3 (((φ [y / x]φ) → x = y) ↔ ((φ ψ) → x = y))
872albii 1357 . 2 (xy((φ [y / x]φ) → x = y) ↔ xy((φ ψ) → x = y))
92, 8bitri 173 1 (∃*xφxy((φ ψ) → x = y))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 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:  mo4  1958  mob2  2715  moop2  3979  dffun4f  4861
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