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Theorem prtlem5 33624
Description: Lemma for prter1 33644, prter2 33646, prter3 33647 and prtex 33645. (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 1840 . 2 𝑣𝑥𝐴 (𝑟𝑥𝑠𝑥)
2 elequ1 1994 . . . . 5 (𝑢 = 𝑟 → (𝑢𝑥𝑟𝑥))
3 elequ1 1994 . . . . 5 (𝑣 = 𝑠 → (𝑣𝑥𝑠𝑥))
42, 3bi2anan9r 917 . . . 4 ((𝑣 = 𝑠𝑢 = 𝑟) → ((𝑢𝑥𝑣𝑥) ↔ (𝑟𝑥𝑠𝑥)))
54rexbidv 3045 . . 3 ((𝑣 = 𝑠𝑢 = 𝑟) → (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
65sbiedv 2409 . 2 (𝑣 = 𝑠 → ([𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
71, 6sbie 2407 1 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  [wsb 1877  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878  df-rex 2913
This theorem is referenced by: (None)
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