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Theorem moi 2692
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
moi (((A 𝐶 B 𝐷) ∃*xφ (ψ χ)) → A = B)
Distinct variable groups:   x,A   x,B   χ,x   ψ,x
Allowed substitution hints:   φ(x)   𝐶(x)   𝐷(x)

Proof of Theorem moi
StepHypRef Expression
1 moi.1 . . . . . 6 (x = A → (φψ))
2 moi.2 . . . . . 6 (x = B → (φχ))
31, 2mob 2691 . . . . 5 (((A 𝐶 B 𝐷) ∃*xφ ψ) → (A = Bχ))
43biimprd 147 . . . 4 (((A 𝐶 B 𝐷) ∃*xφ ψ) → (χA = B))
543expia 1087 . . 3 (((A 𝐶 B 𝐷) ∃*xφ) → (ψ → (χA = B)))
65impd 242 . 2 (((A 𝐶 B 𝐷) ∃*xφ) → ((ψ χ) → A = B))
763impia 1082 1 (((A 𝐶 B 𝐷) ∃*xφ (ψ χ)) → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 867   = wceq 1223   wcel 1366  ∃*wmo 1874
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528
This theorem is referenced by: (None)
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