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Theorem reu7 3031
 Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (x = y → (φψ))
Assertion
Ref Expression
reu7 (∃!x A φ ↔ (x A φ x A y A (ψx = y)))
Distinct variable groups:   x,y,A   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem reu7
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 reu3 3026 . 2 (∃!x A φ ↔ (x A φ z A x A (φx = z)))
2 rmo4.1 . . . . . . 7 (x = y → (φψ))
3 eqeq1 2359 . . . . . . . 8 (x = y → (x = zy = z))
4 eqcom 2355 . . . . . . . 8 (y = zz = y)
53, 4syl6bb 252 . . . . . . 7 (x = y → (x = zz = y))
62, 5imbi12d 311 . . . . . 6 (x = y → ((φx = z) ↔ (ψz = y)))
76cbvralv 2835 . . . . 5 (x A (φx = z) ↔ y A (ψz = y))
87rexbii 2639 . . . 4 (z A x A (φx = z) ↔ z A y A (ψz = y))
9 eqeq1 2359 . . . . . . 7 (z = x → (z = yx = y))
109imbi2d 307 . . . . . 6 (z = x → ((ψz = y) ↔ (ψx = y)))
1110ralbidv 2634 . . . . 5 (z = x → (y A (ψz = y) ↔ y A (ψx = y)))
1211cbvrexv 2836 . . . 4 (z A y A (ψz = y) ↔ x A y A (ψx = y))
138, 12bitri 240 . . 3 (z A x A (φx = z) ↔ x A y A (ψx = y))
1413anbi2i 675 . 2 ((x A φ z A x A (φx = z)) ↔ (x A φ x A y A (ψx = y)))
151, 14bitri 240 1 (∃!x A φ ↔ (x A φ x A y A (ψx = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by: (None)
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