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Theorem eqvincg 2668
Description: A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Assertion
Ref Expression
eqvincg (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem eqvincg
StepHypRef Expression
1 elisset 2568 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 ax-1 5 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐴))
3 eqtr 2057 . . . . . . 7 ((𝑥 = 𝐴𝐴 = 𝐵) → 𝑥 = 𝐵)
43ex 108 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐵))
52, 4jca 290 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
65eximi 1491 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
7 pm3.43 534 . . . . 5 (((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
87eximi 1491 . . . 4 (∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
91, 6, 83syl 17 . . 3 (𝐴𝑉 → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
10 nfv 1421 . . . 4 𝑥 𝐴 = 𝐵
111019.37-1 1564 . . 3 (∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)) → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
129, 11syl 14 . 2 (𝐴𝑉 → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
13 eqtr2 2058 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1413exlimiv 1489 . 2 (∃𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1512, 14impbid1 130 1 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559
This theorem is referenced by:  dff13  5407  f1eqcocnv  5431
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