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Theorem elabgf 2708
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1 𝑥𝐴
elabgf.2 𝑥𝜓
elabgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elabgf (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2 𝑥𝐴
2 nfab1 2196 . . . 4 𝑥{𝑥𝜑}
31, 2nfel 2202 . . 3 𝑥 𝐴 ∈ {𝑥𝜑}
4 elabgf.2 . . 3 𝑥𝜓
53, 4nfbi 1497 . 2 𝑥(𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
6 eleq1 2116 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 elabgf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7bibi12d 228 . 2 (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)))
9 abid 2044 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
101, 5, 8, 9vtoclgf 2629 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wnf 1365  wcel 1409  {cab 2042  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  elabf  2709  elabg  2711  elab3gf  2715  elrabf  2719  bj-intabssel  10315
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