ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clelab GIF version

Theorem clelab 2303
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
clelab (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem clelab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2164 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21anbi2i 457 . . 3 ((𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
32exbii 1605 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
4 df-clel 2173 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
5 nfv 1528 . . 3 𝑦(𝑥 = 𝐴𝜑)
6 nfv 1528 . . . 4 𝑥 𝑦 = 𝐴
7 nfs1v 1939 . . . 4 𝑥[𝑦 / 𝑥]𝜑
86, 7nfan 1565 . . 3 𝑥(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)
9 eqeq1 2184 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
10 sbequ12 1771 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
119, 10anbi12d 473 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
125, 8, 11cbvex 1756 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
133, 4, 123bitr4i 212 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wex 1492  [wsb 1762  wcel 2148  {cab 2163
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by:  elrabi  2890
  Copyright terms: Public domain W3C validator