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Theorem ceqsexOLD 3530
Description: Obsolete version of ceqsex 3529 as of 22-Jan-2025. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ceqsex.1 𝑥𝜓
ceqsex.2 𝐴 ∈ V
ceqsex.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexOLD (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsexOLD
StepHypRef Expression
1 ceqsex.1 . . 3 𝑥𝜓
2 ceqsex.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
32biimpa 476 . . 3 ((𝑥 = 𝐴𝜑) → 𝜓)
41, 3exlimi 2216 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)
52biimprcd 250 . . . 4 (𝜓 → (𝑥 = 𝐴𝜑))
61, 5alrimi 2212 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
7 ceqsex.2 . . . 4 𝐴 ∈ V
87isseti 3497 . . 3 𝑥 𝑥 = 𝐴
9 exintr 1891 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴𝜑)))
106, 8, 9mpisyl 21 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
114, 10impbii 209 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wnf 1782  wcel 2107  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815
This theorem is referenced by: (None)
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