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Theorem exsbOLD 2602
 Description: Obsolete version of exsb 2384 as of 16-Oct-2022. (Contributed by NM, 2-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
exsbOLD (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exsbOLD
StepHypRef Expression
1 nfv 2015 . 2 𝑦𝜑
2 nfa1 2204 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
3 ax12v 2223 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sp 2226 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
63, 5impbid 204 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
71, 2, 6cbvex 2425 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1656  ∃wex 1880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885 This theorem is referenced by: (None)
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