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Theorem rmo4 2728
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmo4.1 (x = y → (φψ))
Assertion
Ref Expression
rmo4 (∃*x A φx 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 rmo4
StepHypRef Expression
1 df-rmo 2308 . 2 (∃*x A φ∃*x(x A φ))
2 an4 520 . . . . . . . . 9 (((x A φ) (y A ψ)) ↔ ((x A y A) (φ ψ)))
3 ancom 253 . . . . . . . . . 10 ((x A y A) ↔ (y A x A))
43anbi1i 431 . . . . . . . . 9 (((x A y A) (φ ψ)) ↔ ((y A x A) (φ ψ)))
52, 4bitri 173 . . . . . . . 8 (((x A φ) (y A ψ)) ↔ ((y A x A) (φ ψ)))
65imbi1i 227 . . . . . . 7 ((((x A φ) (y A ψ)) → x = y) ↔ (((y A x A) (φ ψ)) → x = y))
7 impexp 250 . . . . . . 7 ((((y A x A) (φ ψ)) → x = y) ↔ ((y A x A) → ((φ ψ) → x = y)))
8 impexp 250 . . . . . . 7 (((y A x A) → ((φ ψ) → x = y)) ↔ (y A → (x A → ((φ ψ) → x = y))))
96, 7, 83bitri 195 . . . . . 6 ((((x A φ) (y A ψ)) → x = y) ↔ (y A → (x A → ((φ ψ) → x = y))))
109albii 1356 . . . . 5 (y(((x A φ) (y A ψ)) → x = y) ↔ y(y A → (x A → ((φ ψ) → x = y))))
11 df-ral 2305 . . . . 5 (y A (x A → ((φ ψ) → x = y)) ↔ y(y A → (x A → ((φ ψ) → x = y))))
12 r19.21v 2390 . . . . 5 (y A (x A → ((φ ψ) → x = y)) ↔ (x Ay A ((φ ψ) → x = y)))
1310, 11, 123bitr2i 197 . . . 4 (y(((x A φ) (y A ψ)) → x = y) ↔ (x Ay A ((φ ψ) → x = y)))
1413albii 1356 . . 3 (xy(((x A φ) (y A ψ)) → x = y) ↔ x(x Ay A ((φ ψ) → x = y)))
15 eleq1 2097 . . . . 5 (x = y → (x Ay A))
16 rmo4.1 . . . . 5 (x = y → (φψ))
1715, 16anbi12d 442 . . . 4 (x = y → ((x A φ) ↔ (y A ψ)))
1817mo4 1958 . . 3 (∃*x(x A φ) ↔ xy(((x A φ) (y A ψ)) → x = y))
19 df-ral 2305 . . 3 (x A y A ((φ ψ) → x = y) ↔ x(x Ay A ((φ ψ) → x = y)))
2014, 18, 193bitr4i 201 . 2 (∃*x(x A φ) ↔ x A y A ((φ ψ) → x = y))
211, 20bitri 173 1 (∃*x A φx A y A ((φ ψ) → x = y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  ∃*wmo 1898  wral 2300  ∃*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-rmo 2308
This theorem is referenced by:  reu4  2729
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