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Theorem mob 2717
Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1 (x = A → (φψ))
moi.2 (x = B → (φχ))
Assertion
Ref Expression
mob (((A 𝐶 B 𝐷) ∃*xφ ψ) → (A = Bχ))
Distinct variable groups:   x,A   x,B   χ,x   ψ,x
Allowed substitution hints:   φ(x)   𝐶(x)   𝐷(x)

Proof of Theorem mob
StepHypRef Expression
1 elex 2560 . . . . 5 (B 𝐷B V)
2 nfcv 2175 . . . . . . . 8 xA
3 nfv 1418 . . . . . . . . . 10 x B V
4 nfmo1 1909 . . . . . . . . . 10 x∃*xφ
5 nfv 1418 . . . . . . . . . 10 xψ
63, 4, 5nf3an 1455 . . . . . . . . 9 x(B V ∃*xφ ψ)
7 nfv 1418 . . . . . . . . 9 x(A = Bχ)
86, 7nfim 1461 . . . . . . . 8 x((B V ∃*xφ ψ) → (A = Bχ))
9 moi.1 . . . . . . . . . 10 (x = A → (φψ))
1093anbi3d 1212 . . . . . . . . 9 (x = A → ((B V ∃*xφ φ) ↔ (B V ∃*xφ ψ)))
11 eqeq1 2043 . . . . . . . . . 10 (x = A → (x = BA = B))
1211bibi1d 222 . . . . . . . . 9 (x = A → ((x = Bχ) ↔ (A = Bχ)))
1310, 12imbi12d 223 . . . . . . . 8 (x = A → (((B V ∃*xφ φ) → (x = Bχ)) ↔ ((B V ∃*xφ ψ) → (A = Bχ))))
14 moi.2 . . . . . . . . 9 (x = B → (φχ))
1514mob2 2715 . . . . . . . 8 ((B V ∃*xφ φ) → (x = Bχ))
162, 8, 13, 15vtoclgf 2606 . . . . . . 7 (A 𝐶 → ((B V ∃*xφ ψ) → (A = Bχ)))
1716com12 27 . . . . . 6 ((B V ∃*xφ ψ) → (A 𝐶 → (A = Bχ)))
18173expib 1106 . . . . 5 (B V → ((∃*xφ ψ) → (A 𝐶 → (A = Bχ))))
191, 18syl 14 . . . 4 (B 𝐷 → ((∃*xφ ψ) → (A 𝐶 → (A = Bχ))))
2019com3r 73 . . 3 (A 𝐶 → (B 𝐷 → ((∃*xφ ψ) → (A = Bχ))))
2120imp 115 . 2 ((A 𝐶 B 𝐷) → ((∃*xφ ψ) → (A = Bχ)))
22213impib 1101 1 (((A 𝐶 B 𝐷) ∃*xφ ψ) → (A = Bχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃*wmo 1898  Vcvv 2551
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  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  moi  2718  rmob  2844
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