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Theorem cbvexeqsetf 3493
Description: The expression 𝑥𝑥 = 𝐴 means "𝐴 is a set" even if 𝐴 contains 𝑥 as a bound variable. This lemma helps minimizing axiom or df-clab 2713 usage in some cases. Extracted from the proof of issetft 3494. (Contributed by Wolf Lammen, 30-Jul-2025.)
Assertion
Ref Expression
cbvexeqsetf (𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem cbvexeqsetf
StepHypRef Expression
1 nfnfc1 2906 . . . 4 𝑥𝑥𝐴
2 nfv 1912 . . . 4 𝑦𝑥𝐴
3 nfvd 1913 . . . 4 (𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
4 nfcvd 2904 . . . . . 6 (𝑥𝐴𝑥𝑦)
5 id 22 . . . . . 6 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2914 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
76nfnd 1856 . . . 4 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8 eqeq1 2739 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
98notbid 318 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
109a1i 11 . . . 4 (𝑥𝐴 → (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴)))
111, 2, 3, 7, 10cbv2w 2338 . . 3 (𝑥𝐴 → (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑦 ¬ 𝑦 = 𝐴))
12 alnex 1778 . . 3 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
13 alnex 1778 . . 3 (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
1411, 12, 133bitr3g 313 . 2 (𝑥𝐴 → (¬ ∃𝑥 𝑥 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴))
1514con4bid 317 1 (𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-cleq 2727  df-nfc 2890
This theorem is referenced by:  issetft  3494  spcimgft  3546
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