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Theorem prtlem5 34645
Description: Lemma for prter1 34664, prter2 34666, prter3 34667 and prtex 34665. (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 2005 . 2 𝑣𝑥𝐴 (𝑟𝑥𝑠𝑥)
2 elequ1 2164 . . . . 5 (𝑢 = 𝑟 → (𝑢𝑥𝑟𝑥))
3 elequ1 2164 . . . . 5 (𝑣 = 𝑠 → (𝑣𝑥𝑠𝑥))
42, 3bi2anan9r 623 . . . 4 ((𝑣 = 𝑠𝑢 = 𝑟) → ((𝑢𝑥𝑣𝑥) ↔ (𝑟𝑥𝑠𝑥)))
54rexbidv 3251 . . 3 ((𝑣 = 𝑠𝑢 = 𝑟) → (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
65sbiedv 2571 . 2 (𝑣 = 𝑠 → ([𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
71, 6sbie 2569 1 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  [wsb 2061  wrex 3108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-10 2186  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864  df-sb 2062  df-rex 3113
This theorem is referenced by: (None)
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