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Theorem eqvincg 2645
 Description: A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Assertion
Ref Expression
eqvincg (A 𝑉 → (A = Bx(x = A x = B)))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝑉(x)

Proof of Theorem eqvincg
StepHypRef Expression
1 elisset 2545 . . . 4 (A 𝑉x x = A)
2 ax-1 5 . . . . . 6 (x = A → (A = Bx = A))
3 eqtr 2039 . . . . . . 7 ((x = A A = B) → x = B)
43ex 108 . . . . . 6 (x = A → (A = Bx = B))
52, 4jca 290 . . . . 5 (x = A → ((A = Bx = A) (A = Bx = B)))
65eximi 1473 . . . 4 (x x = Ax((A = Bx = A) (A = Bx = B)))
7 pm3.43 521 . . . . 5 (((A = Bx = A) (A = Bx = B)) → (A = B → (x = A x = B)))
87eximi 1473 . . . 4 (x((A = Bx = A) (A = Bx = B)) → x(A = B → (x = A x = B)))
91, 6, 83syl 17 . . 3 (A 𝑉x(A = B → (x = A x = B)))
10 nfv 1402 . . . 4 x A = B
111019.37-1 1546 . . 3 (x(A = B → (x = A x = B)) → (A = Bx(x = A x = B)))
129, 11syl 14 . 2 (A 𝑉 → (A = Bx(x = A x = B)))
13 eqtr2 2040 . . 3 ((x = A x = B) → A = B)
1413exlimiv 1471 . 2 (x(x = A x = B) → A = B)
1512, 14impbid1 130 1 (A 𝑉 → (A = Bx(x = A x = B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374 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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537 This theorem is referenced by:  dff13  5332  f1eqcocnv  5356
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