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Theorem iineq1 3662
 Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iineq1 (A = B x A 𝐶 = x B 𝐶)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem iineq1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 raleq 2499 . . 3 (A = B → (x A y 𝐶x B y 𝐶))
21abbidv 2152 . 2 (A = B → {yx A y 𝐶} = {yx B y 𝐶})
3 df-iin 3651 . 2 x A 𝐶 = {yx A y 𝐶}
4 df-iin 3651 . 2 x B 𝐶 = {yx B y 𝐶}
52, 3, 43eqtr4g 2094 1 (A = B x A 𝐶 = x B 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∩ ciin 3649 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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-iin 3651 This theorem is referenced by:  riin0  3719  iin0r  3913
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