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Theorem reu4 2729
 Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
Hypothesis
Ref Expression
rmo4.1 (x = y → (φψ))
Assertion
Ref Expression
reu4 (∃!x A φ ↔ (x A φ x A y A ((φ ψ) → x = y)))
Distinct variable groups:   x,y,A   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem reu4
StepHypRef Expression
1 reu5 2516 . 2 (∃!x A φ ↔ (x A φ ∃*x A φ))
2 rmo4.1 . . . 4 (x = y → (φψ))
32rmo4 2728 . . 3 (∃*x A φx A y A ((φ ψ) → x = y))
43anbi2i 430 . 2 ((x A φ ∃*x A φ) ↔ (x A φ x A y A ((φ ψ) → x = y)))
51, 4bitri 173 1 (∃!x A φ ↔ (x A φ x A y A ((φ ψ) → x = y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302  ∃*wrmo 2303 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-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308 This theorem is referenced by:  reuind  2738  receuap  7412  cju  7674
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