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Mirrors > Home > NFE Home > Th. List > elab2 | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elab2.1 | ⊢ A ∈ V |
elab2.2 | ⊢ (x = A → (φ ↔ ψ)) |
elab2.3 | ⊢ B = {x ∣ φ} |
Ref | Expression |
---|---|
elab2 | ⊢ (A ∈ B ↔ ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab2.1 | . 2 ⊢ A ∈ V | |
2 | elab2.2 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
3 | elab2.3 | . . 3 ⊢ B = {x ∣ φ} | |
4 | 2, 3 | elab2g 2987 | . 2 ⊢ (A ∈ V → (A ∈ B ↔ ψ)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (A ∈ B ↔ ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 {cab 2339 Vcvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 |
This theorem is referenced by: elpw 3728 elint 3932 opkelopkabg 4245 0ceven 4505 eventfin 4517 oddtfin 4518 dfphi2 4569 phi11lem1 4595 0cnelphi 4597 proj1op 4600 proj2op 4601 opabid 4695 oprabid 5550 elfuns 5829 |
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