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Theorem rmo3 2826
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 2292 . 2 (∃*x A φ∃*x(x A φ))
2 sban 1811 . . . . . . . . . . 11 ([y / x](x A φ) ↔ ([y / x]x A [y / x]φ))
3 clelsb3 2124 . . . . . . . . . . . 12 ([y / x]x Ay A)
43anbi1i 434 . . . . . . . . . . 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 433 . . . . . . . . 9 (((x A φ) [y / x](x A φ)) ↔ ((x A φ) (y A [y / x]φ)))
7 an4 507 . . . . . . . . 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 434 . . . . . . . . 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 1339 . . . . 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 2289 . . . . 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 2374 . . . . 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 1339 . . 3 (xy(((x A φ) [y / x](x A φ)) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
20 nfv 1402 . . . . 5 y x A
21 rmo2.1 . . . . 5 yφ
2220, 21nfan 1439 . . . 4 y(x A φ)
2322mo3 1936 . . 3 (∃*x(x A φ) ↔ xy(((x A φ) [y / x](x A φ)) → x = y))
24 df-ral 2289 . . 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 1226  wnf 1329   wcel 1374  [wsb 1627  ∃*wmo 1883  wral 2284  ∃*wrmo 2287
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-cleq 2015  df-clel 2018  df-ral 2289  df-rmo 2292
This theorem is referenced by: (None)
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