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Theorem prtlem5 34667
 Description: Lemma for prter1 34686, prter2 34688, prter3 34689 and prtex 34687. (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 1992 . 2 𝑣𝑥𝐴 (𝑟𝑥𝑠𝑥)
2 elequ1 2146 . . . . 5 (𝑢 = 𝑟 → (𝑢𝑥𝑟𝑥))
3 elequ1 2146 . . . . 5 (𝑣 = 𝑠 → (𝑣𝑥𝑠𝑥))
42, 3bi2anan9r 954 . . . 4 ((𝑣 = 𝑠𝑢 = 𝑟) → ((𝑢𝑥𝑣𝑥) ↔ (𝑟𝑥𝑠𝑥)))
54rexbidv 3190 . . 3 ((𝑣 = 𝑠𝑢 = 𝑟) → (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
65sbiedv 2547 . 2 (𝑣 = 𝑠 → ([𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥)))
71, 6sbie 2545 1 ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383  [wsb 2046  ∃wrex 3051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-10 2168  ax-12 2196  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047  df-rex 3056 This theorem is referenced by: (None)
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