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Theorem rmo3 3133
 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 2622 . 2 (∃*x A φ∃*x(x A φ))
2 sban 2069 . . . . . . . . . . 11 ([y / x](x A φ) ↔ ([y / x]x A [y / x]φ))
3 clelsb3 2455 . . . . . . . . . . . 12 ([y / x]x Ay A)
43anbi1i 676 . . . . . . . . . . 11 (([y / x]x A [y / x]φ) ↔ (y A [y / x]φ))
52, 4bitri 240 . . . . . . . . . 10 ([y / x](x A φ) ↔ (y A [y / x]φ))
65anbi2i 675 . . . . . . . . 9 (((x A φ) [y / x](x A φ)) ↔ ((x A φ) (y A [y / x]φ)))
7 an4 797 . . . . . . . . 9 (((x A φ) (y A [y / x]φ)) ↔ ((x A y A) (φ [y / x]φ)))
8 ancom 437 . . . . . . . . . 10 ((x A y A) ↔ (y A x A))
98anbi1i 676 . . . . . . . . 9 (((x A y A) (φ [y / x]φ)) ↔ ((y A x A) (φ [y / x]φ)))
106, 7, 93bitri 262 . . . . . . . 8 (((x A φ) [y / x](x A φ)) ↔ ((y A x A) (φ [y / x]φ)))
1110imbi1i 315 . . . . . . 7 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (((y A x A) (φ [y / x]φ)) → x = y))
12 impexp 433 . . . . . . 7 ((((y A x A) (φ [y / x]φ)) → x = y) ↔ ((y A x A) → ((φ [y / x]φ) → x = y)))
13 impexp 433 . . . . . . 7 (((y A x A) → ((φ [y / x]φ) → x = y)) ↔ (y A → (x A → ((φ [y / x]φ) → x = y))))
1411, 12, 133bitri 262 . . . . . 6 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (y A → (x A → ((φ [y / x]φ) → x = y))))
1514albii 1566 . . . . 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 2619 . . . . 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 2701 . . . . 5 (y A (x A → ((φ [y / x]φ) → x = y)) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
1815, 16, 173bitr2i 264 . . . 4 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
1918albii 1566 . . 3 (xy(((x A φ) [y / x](x A φ)) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
20 nfv 1619 . . . . 5 y x A
21 rmo2.1 . . . . 5 yφ
2220, 21nfan 1824 . . . 4 y(x A φ)
2322mo3 2235 . . 3 (∃*x(x A φ) ↔ xy(((x A φ) [y / x](x A φ)) → x = y))
24 df-ral 2619 . . 3 (x A y A ((φ [y / x]φ) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
2519, 23, 243bitr4i 268 . 2 (∃*x(x A φ) ↔ x A y A ((φ [y / x]φ) → x = y))
261, 25bitri 240 1 (∃*x A φx A y A ((φ [y / x]φ) → x = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ral 2619  df-rmo 2622 This theorem is referenced by: (None)
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