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Theorem rmo3 2843
 Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1 yφ
Assertion
Ref Expression
rmo3 (∃*x A φx A y A ((φ [y / x]φ) → x = y))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo3
StepHypRef Expression
1 df-rmo 2308 . 2 (∃*x A φ∃*x(x A φ))
2 sban 1826 . . . . . . . . . . 11 ([y / x](x A φ) ↔ ([y / x]x A [y / x]φ))
3 clelsb3 2139 . . . . . . . . . . . 12 ([y / x]x Ay A)
43anbi1i 431 . . . . . . . . . . 11 (([y / x]x A [y / x]φ) ↔ (y A [y / x]φ))
52, 4bitri 173 . . . . . . . . . 10 ([y / x](x A φ) ↔ (y A [y / x]φ))
65anbi2i 430 . . . . . . . . 9 (((x A φ) [y / x](x A φ)) ↔ ((x A φ) (y A [y / x]φ)))
7 an4 520 . . . . . . . . 9 (((x A φ) (y A [y / x]φ)) ↔ ((x A y A) (φ [y / x]φ)))
8 ancom 253 . . . . . . . . . 10 ((x A y A) ↔ (y A x A))
98anbi1i 431 . . . . . . . . 9 (((x A y A) (φ [y / x]φ)) ↔ ((y A x A) (φ [y / x]φ)))
106, 7, 93bitri 195 . . . . . . . 8 (((x A φ) [y / x](x A φ)) ↔ ((y A x A) (φ [y / x]φ)))
1110imbi1i 227 . . . . . . 7 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (((y A x A) (φ [y / x]φ)) → x = y))
12 impexp 250 . . . . . . 7 ((((y A x A) (φ [y / x]φ)) → x = y) ↔ ((y A x A) → ((φ [y / x]φ) → x = y)))
13 impexp 250 . . . . . . 7 (((y A x A) → ((φ [y / x]φ) → x = y)) ↔ (y A → (x A → ((φ [y / x]φ) → x = y))))
1411, 12, 133bitri 195 . . . . . 6 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (y A → (x A → ((φ [y / x]φ) → x = y))))
1514albii 1356 . . . . 5 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ y(y A → (x A → ((φ [y / x]φ) → x = y))))
16 df-ral 2305 . . . . 5 (y A (x A → ((φ [y / x]φ) → x = y)) ↔ y(y A → (x A → ((φ [y / x]φ) → x = y))))
17 r19.21v 2390 . . . . 5 (y A (x A → ((φ [y / x]φ) → x = y)) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
1815, 16, 173bitr2i 197 . . . 4 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
1918albii 1356 . . 3 (xy(((x A φ) [y / x](x A φ)) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
20 nfv 1418 . . . . 5 y x A
21 rmo2.1 . . . . 5 yφ
2220, 21nfan 1454 . . . 4 y(x A φ)
2322mo3 1951 . . 3 (∃*x(x A φ) ↔ xy(((x A φ) [y / x](x A φ)) → x = y))
24 df-ral 2305 . . 3 (x A y A ((φ [y / x]φ) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
2519, 23, 243bitr4i 201 . 2 (∃*x(x A φ) ↔ x A y A ((φ [y / x]φ) → x = y))
261, 25bitri 173 1 (∃*x A φx A y A ((φ [y / x]φ) → x = y))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346   ∈ wcel 1390  [wsb 1642  ∃*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-bndl 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: (None)
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