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Theorem rexsb 44263
Description: An equivalent expression for restricted existence, analogous to exsb 2358. (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 1922 . 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 3350 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-10 2141  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067
This theorem is referenced by:  rexrsb  44264  2rexsb  44265
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