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Theorem eqvincg 2873
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 2763 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 ax-1 6 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐴))
3 eqtr 2205 . . . . . . 7 ((𝑥 = 𝐴𝐴 = 𝐵) → 𝑥 = 𝐵)
43ex 115 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐵))
52, 4jca 306 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
65eximi 1610 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
7 pm3.43 602 . . . . 5 (((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
87eximi 1610 . . . 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 1538 . . . 4 𝑥 𝐴 = 𝐵
111019.37-1 1684 . . 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 2206 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1413exlimiv 1608 . 2 (∃𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1512, 14impbid1 142 1 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wex 1502  wcel 2158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751
This theorem is referenced by:  dff13  5782  f1eqcocnv  5805
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