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Theorem rexsb 43715
 Description: An equivalent expression for restricted existence, analogous to exsb 2367. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
rexsb (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsb
StepHypRef Expression
1 nfv 1915 . 2 𝑦𝜑
2 nfa1 2152 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
3 ax12v 2176 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sp 2180 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
63, 5impbid 215 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
71, 2, 6cbvrexw 3388 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112 This theorem is referenced by:  rexrsb  43716  2rexsb  43717
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