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Theorem moeq3 3013
 Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)
Hypotheses
Ref Expression
moeq3.1 B V
moeq3.2 C V
moeq3.3 ¬ (φ ψ)
Assertion
Ref Expression
moeq3 ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))
Distinct variable groups:   φ,x   ψ,x   x,A   x,B   x,C

Proof of Theorem moeq3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2362 . . . . . . 7 (y = A → (x = yx = A))
21anbi2d 684 . . . . . 6 (y = A → ((φ x = y) ↔ (φ x = A)))
3 biidd 228 . . . . . 6 (y = A → ((¬ (φ ψ) x = B) ↔ (¬ (φ ψ) x = B)))
4 biidd 228 . . . . . 6 (y = A → ((ψ x = C) ↔ (ψ x = C)))
52, 3, 43orbi123d 1251 . . . . 5 (y = A → (((φ x = y) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
65eubidv 2212 . . . 4 (y = A → (∃!x((φ x = y) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
7 vex 2862 . . . . 5 y V
8 moeq3.1 . . . . 5 B V
9 moeq3.2 . . . . 5 C V
10 moeq3.3 . . . . 5 ¬ (φ ψ)
117, 8, 9, 10eueq3 3011 . . . 4 ∃!x((φ x = y) (¬ (φ ψ) x = B) (ψ x = C))
126, 11vtoclg 2914 . . 3 (A V → ∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
13 eumo 2244 . . 3 (∃!x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) → ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
1412, 13syl 15 . 2 (A V → ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
15 vex 2862 . . . . . . . . 9 x V
16 eleq1 2413 . . . . . . . . 9 (x = A → (x V ↔ A V))
1715, 16mpbii 202 . . . . . . . 8 (x = AA V)
18 pm2.21 100 . . . . . . . 8 A V → (A V → x = y))
1917, 18syl5 28 . . . . . . 7 A V → (x = Ax = y))
2019anim2d 548 . . . . . 6 A V → ((φ x = A) → (φ x = y)))
2120orim1d 812 . . . . 5 A V → (((φ x = A) ((¬ (φ ψ) x = B) (ψ x = C))) → ((φ x = y) ((¬ (φ ψ) x = B) (ψ x = C)))))
22 3orass 937 . . . . 5 (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ((φ x = A) ((¬ (φ ψ) x = B) (ψ x = C))))
23 3orass 937 . . . . 5 (((φ x = y) (¬ (φ ψ) x = B) (ψ x = C)) ↔ ((φ x = y) ((¬ (φ ψ) x = B) (ψ x = C))))
2421, 22, 233imtr4g 261 . . . 4 A V → (((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) → ((φ x = y) (¬ (φ ψ) x = B) (ψ x = C))))
2524alrimiv 1631 . . 3 A V → x(((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) → ((φ x = y) (¬ (φ ψ) x = B) (ψ x = C))))
26 euimmo 2253 . . 3 (x(((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)) → ((φ x = y) (¬ (φ ψ) x = B) (ψ x = C))) → (∃!x((φ x = y) (¬ (φ ψ) x = B) (ψ x = C)) → ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))))
2725, 11, 26ee10 1376 . 2 A V → ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C)))
2814, 27pm2.61i 156 1 ∃*x((φ x = A) (¬ (φ ψ) x = B) (ψ x = C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∨ w3o 933  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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