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Theorem rmo4 3029
 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 2622 . 2 (∃*x A φ∃*x(x A φ))
2 an4 797 . . . . . . . . 9 (((x A φ) (y A ψ)) ↔ ((x A y A) (φ ψ)))
3 ancom 437 . . . . . . . . . 10 ((x A y A) ↔ (y A x A))
43anbi1i 676 . . . . . . . . 9 (((x A y A) (φ ψ)) ↔ ((y A x A) (φ ψ)))
52, 4bitri 240 . . . . . . . 8 (((x A φ) (y A ψ)) ↔ ((y A x A) (φ ψ)))
65imbi1i 315 . . . . . . 7 ((((x A φ) (y A ψ)) → x = y) ↔ (((y A x A) (φ ψ)) → x = y))
7 impexp 433 . . . . . . 7 ((((y A x A) (φ ψ)) → x = y) ↔ ((y A x A) → ((φ ψ) → x = y)))
8 impexp 433 . . . . . . 7 (((y A x A) → ((φ ψ) → x = y)) ↔ (y A → (x A → ((φ ψ) → x = y))))
96, 7, 83bitri 262 . . . . . 6 ((((x A φ) (y A ψ)) → x = y) ↔ (y A → (x A → ((φ ψ) → x = y))))
109albii 1566 . . . . 5 (y(((x A φ) (y A ψ)) → x = y) ↔ y(y A → (x A → ((φ ψ) → x = y))))
11 df-ral 2619 . . . . 5 (y A (x A → ((φ ψ) → x = y)) ↔ y(y A → (x A → ((φ ψ) → x = y))))
12 r19.21v 2701 . . . . 5 (y A (x A → ((φ ψ) → x = y)) ↔ (x Ay A ((φ ψ) → x = y)))
1310, 11, 123bitr2i 264 . . . 4 (y(((x A φ) (y A ψ)) → x = y) ↔ (x Ay A ((φ ψ) → x = y)))
1413albii 1566 . . 3 (xy(((x A φ) (y A ψ)) → x = y) ↔ x(x Ay A ((φ ψ) → x = y)))
15 eleq1 2413 . . . . 5 (x = y → (x Ay A))
16 rmo4.1 . . . . 5 (x = y → (φψ))
1715, 16anbi12d 691 . . . 4 (x = y → ((x A φ) ↔ (y A ψ)))
1817mo4 2237 . . 3 (∃*x(x A φ) ↔ xy(((x A φ) (y A ψ)) → x = y))
19 df-ral 2619 . . 3 (x A y A ((φ ψ) → x = y) ↔ x(x Ay A ((φ ψ) → x = y)))
2014, 18, 193bitr4i 268 . 2 (∃*x(x A φ) ↔ x A y A ((φ ψ) → x = y))
211, 20bitri 240 1 (∃*x A φx A y A ((φ ψ) → x = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ 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:  reu4  3030
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