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Theorem clel4g 3560
Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2179. (Revised by BJ, 1-Sep-2024.)
Assertion
Ref Expression
clel4g (𝐵𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel4g
StepHypRef Expression
1 elisset 2814 . . . 4 (𝐵𝑉 → ∃𝑥 𝑥 = 𝐵)
2 biimt 364 . . . 4 (∃𝑥 𝑥 = 𝐵 → (𝐴𝐵 ↔ (∃𝑥 𝑥 = 𝐵𝐴𝐵)))
31, 2syl 17 . . 3 (𝐵𝑉 → (𝐴𝐵 ↔ (∃𝑥 𝑥 = 𝐵𝐴𝐵)))
4 19.23v 1949 . . 3 (∀𝑥(𝑥 = 𝐵𝐴𝐵) ↔ (∃𝑥 𝑥 = 𝐵𝐴𝐵))
53, 4bitr4di 292 . 2 (𝐵𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝐵)))
6 eleq2 2821 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
76bicomd 226 . . . 4 (𝑥 = 𝐵 → (𝐴𝐵𝐴𝑥))
87pm5.74i 274 . . 3 ((𝑥 = 𝐵𝐴𝐵) ↔ (𝑥 = 𝐵𝐴𝑥))
98albii 1826 . 2 (∀𝑥(𝑥 = 𝐵𝐴𝐵) ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥))
105, 9bitrdi 290 1 (𝐵𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐵𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540   = wceq 1542  wex 1786  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811
This theorem is referenced by:  clel4  3561  intprg  4869
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