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Theorem eqvincg 29154
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 3206 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 ax-1 6 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐴))
3 eqtr 2645 . . . . . . 7 ((𝑥 = 𝐴𝐴 = 𝐵) → 𝑥 = 𝐵)
43ex 450 . . . . . 6 (𝑥 = 𝐴 → (𝐴 = 𝐵𝑥 = 𝐵))
52, 4jca 554 . . . . 5 (𝑥 = 𝐴 → ((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
65eximi 1759 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)))
7 pm3.43 905 . . . . 5 (((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → (𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
87eximi 1759 . . . 4 (∃𝑥((𝐴 = 𝐵𝑥 = 𝐴) ∧ (𝐴 = 𝐵𝑥 = 𝐵)) → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
91, 6, 83syl 18 . . 3 (𝐴𝑉 → ∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)))
10 19.37v 1912 . . 3 (∃𝑥(𝐴 = 𝐵 → (𝑥 = 𝐴𝑥 = 𝐵)) ↔ (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
119, 10sylib 208 . 2 (𝐴𝑉 → (𝐴 = 𝐵 → ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
12 eqtr2 2646 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1312exlimiv 1860 . 2 (∃𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
1411, 13impbid1 215 1 (𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-12 2049  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1483  df-ex 1702  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-v 3193
This theorem is referenced by:  funcnv5mpt  29303
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