New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  2reu5lem3 GIF version

Theorem 2reu5lem3 3043
 Description: Lemma for 2reu5 3044. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3131. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5lem3 ((∃!x A ∃!y B φ x A ∃*y B φ) ↔ (x A y B φ zwx A y B (φ → (x = z y = w))))
Distinct variable groups:   y,w,z,A   x,w,B,z   x,y   φ,w,z
Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem 2reu5lem3
StepHypRef Expression
1 2reu5lem1 3041 . . 3 (∃!x A ∃!y B φ∃!x∃!y(x A y B φ))
2 2reu5lem2 3042 . . 3 (x A ∃*y B φx∃*y(x A y B φ))
31, 2anbi12i 678 . 2 ((∃!x A ∃!y B φ x A ∃*y B φ) ↔ (∃!x∃!y(x A y B φ) x∃*y(x A y B φ)))
4 2eu5 2288 . 2 ((∃!x∃!y(x A y B φ) x∃*y(x A y B φ)) ↔ (xy(x A y B φ) zwxy((x A y B φ) → (x = z y = w))))
5 3anass 938 . . . . . . 7 ((x A y B φ) ↔ (x A (y B φ)))
65exbii 1582 . . . . . 6 (y(x A y B φ) ↔ y(x A (y B φ)))
7 19.42v 1905 . . . . . 6 (y(x A (y B φ)) ↔ (x A y(y B φ)))
8 df-rex 2620 . . . . . . . 8 (y B φy(y B φ))
98bicomi 193 . . . . . . 7 (y(y B φ) ↔ y B φ)
109anbi2i 675 . . . . . 6 ((x A y(y B φ)) ↔ (x A y B φ))
116, 7, 103bitri 262 . . . . 5 (y(x A y B φ) ↔ (x A y B φ))
1211exbii 1582 . . . 4 (xy(x A y B φ) ↔ x(x A y B φ))
13 df-rex 2620 . . . 4 (x A y B φx(x A y B φ))
1412, 13bitr4i 243 . . 3 (xy(x A y B φ) ↔ x A y B φ)
15 3anan12 947 . . . . . . . . . . 11 ((x A y B φ) ↔ (y B (x A φ)))
1615imbi1i 315 . . . . . . . . . 10 (((x A y B φ) → (x = z y = w)) ↔ ((y B (x A φ)) → (x = z y = w)))
17 impexp 433 . . . . . . . . . 10 (((y B (x A φ)) → (x = z y = w)) ↔ (y B → ((x A φ) → (x = z y = w))))
18 impexp 433 . . . . . . . . . . 11 (((x A φ) → (x = z y = w)) ↔ (x A → (φ → (x = z y = w))))
1918imbi2i 303 . . . . . . . . . 10 ((y B → ((x A φ) → (x = z y = w))) ↔ (y B → (x A → (φ → (x = z y = w)))))
2016, 17, 193bitri 262 . . . . . . . . 9 (((x A y B φ) → (x = z y = w)) ↔ (y B → (x A → (φ → (x = z y = w)))))
2120albii 1566 . . . . . . . 8 (y((x A y B φ) → (x = z y = w)) ↔ y(y B → (x A → (φ → (x = z y = w)))))
22 df-ral 2619 . . . . . . . 8 (y B (x A → (φ → (x = z y = w))) ↔ y(y B → (x A → (φ → (x = z y = w)))))
23 r19.21v 2701 . . . . . . . 8 (y B (x A → (φ → (x = z y = w))) ↔ (x Ay B (φ → (x = z y = w))))
2421, 22, 233bitr2i 264 . . . . . . 7 (y((x A y B φ) → (x = z y = w)) ↔ (x Ay B (φ → (x = z y = w))))
2524albii 1566 . . . . . 6 (xy((x A y B φ) → (x = z y = w)) ↔ x(x Ay B (φ → (x = z y = w))))
26 df-ral 2619 . . . . . 6 (x A y B (φ → (x = z y = w)) ↔ x(x Ay B (φ → (x = z y = w))))
2725, 26bitr4i 243 . . . . 5 (xy((x A y B φ) → (x = z y = w)) ↔ x A y B (φ → (x = z y = w)))
2827exbii 1582 . . . 4 (wxy((x A y B φ) → (x = z y = w)) ↔ wx A y B (φ → (x = z y = w)))
2928exbii 1582 . . 3 (zwxy((x A y B φ) → (x = z y = w)) ↔ zwx A y B (φ → (x = z y = w)))
3014, 29anbi12i 678 . 2 ((xy(x A y B φ) zwxy((x A y B φ) → (x = z y = w))) ↔ (x A y B φ zwx A y B (φ → (x = z y = w))))
313, 4, 303bitri 262 1 ((∃!x A ∃!y B φ x A ∃*y B φ) ↔ (x A y B φ zwx A y B (φ → (x = z y = w))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  2reu5  3044
 Copyright terms: Public domain W3C validator