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Mirrors > Home > NFE Home > Th. List > abbi2dv | GIF version |
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) |
Ref | Expression |
---|---|
abbirdv.1 | ⊢ (φ → (x ∈ A ↔ ψ)) |
Ref | Expression |
---|---|
abbi2dv | ⊢ (φ → A = {x ∣ ψ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbirdv.1 | . . 3 ⊢ (φ → (x ∈ A ↔ ψ)) | |
2 | 1 | alrimiv 1631 | . 2 ⊢ (φ → ∀x(x ∈ A ↔ ψ)) |
3 | abeq2 2458 | . 2 ⊢ (A = {x ∣ ψ} ↔ ∀x(x ∈ A ↔ ψ)) | |
4 | 2, 3 | sylibr 203 | 1 ⊢ (φ → A = {x ∣ ψ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 {cab 2339 |
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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: sbab 2475 iftrue 3668 iffalse 3669 phialllem1 4616 isoini 5497 |
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