New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  r19.12 GIF version

Theorem r19.12 2727
 Description: Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.12 (x A y B φy B x A φ)
Distinct variable groups:   x,y   y,A   x,B
Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem r19.12
StepHypRef Expression
1 nfcv 2489 . . . 4 yA
2 nfra1 2664 . . . 4 yy B φ
31, 2nfrex 2669 . . 3 yx A y B φ
4 ax-1 5 . . 3 (x A y B φ → (y Bx A y B φ))
53, 4ralrimi 2695 . 2 (x A y B φy B x A y B φ)
6 rsp 2674 . . . . 5 (y B φ → (y Bφ))
76com12 27 . . . 4 (y B → (y B φφ))
87reximdv 2725 . . 3 (y B → (x A y B φx A φ))
98ralimia 2687 . 2 (y B x A y B φy B x A φ)
105, 9syl 15 1 (x A y B φy B x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620 This theorem is referenced by:  iuniin  3979
 Copyright terms: Public domain W3C validator