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Theorem prtlem5 35876
Description: Lemma for prter1 35895, prter2 35897, prter3 35898 and prtex 35896. (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 elequ1 2112 . . . 4 (𝑢 = 𝑟 → (𝑢𝑥𝑟𝑥))
2 elequ1 2112 . . . 4 (𝑣 = 𝑠 → (𝑣𝑥𝑠𝑥))
31, 2bi2anan9r 636 . . 3 ((𝑣 = 𝑠𝑢 = 𝑟) → ((𝑢𝑥𝑣𝑥) ↔ (𝑟𝑥𝑠𝑥)))
43rexbidv 3294 . 2 ((𝑣 = 𝑠𝑢 = 𝑟) → (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
542sbievw 2096 1 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  [wsb 2060  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-rex 3141
This theorem is referenced by: (None)
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