Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem5 Structured version   Visualization version   GIF version

Theorem prtlem5 35527
Description: Lemma for prter1 35546, prter2 35548, prter3 35549 and prtex 35547. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
prtlem5 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑟   𝑢,𝑠,𝑣,𝑥   𝑢,𝐴,𝑣,𝑥
Allowed substitution hints:   𝐴(𝑠,𝑟)

Proof of Theorem prtlem5
StepHypRef Expression
1 nfv 1892 . 2 𝑣𝑥𝐴 (𝑟𝑥𝑠𝑥)
2 elequ1 2088 . . . . 5 (𝑢 = 𝑟 → (𝑢𝑥𝑟𝑥))
3 elequ1 2088 . . . . 5 (𝑣 = 𝑠 → (𝑣𝑥𝑠𝑥))
42, 3bi2anan9r 636 . . . 4 ((𝑣 = 𝑠𝑢 = 𝑟) → ((𝑢𝑥𝑣𝑥) ↔ (𝑟𝑥𝑠𝑥)))
54rexbidv 3260 . . 3 ((𝑣 = 𝑠𝑢 = 𝑟) → (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
65sbiedv 2500 . 2 (𝑣 = 𝑠 → ([𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
71, 6sbie 2498 1 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  [wsb 2042  wrex 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-10 2112  ax-12 2141  ax-13 2344
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043  df-rex 3111
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator