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Mirrors > Home > NFE Home > Th. List > ss2ab | GIF version |
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.) |
Ref | Expression |
---|---|
ss2ab | ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2491 | . . 3 ⊢ Ⅎx{x ∣ φ} | |
2 | nfab1 2491 | . . 3 ⊢ Ⅎx{x ∣ ψ} | |
3 | 1, 2 | dfss2f 3264 | . 2 ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(x ∈ {x ∣ φ} → x ∈ {x ∣ ψ})) |
4 | abid 2341 | . . . 4 ⊢ (x ∈ {x ∣ φ} ↔ φ) | |
5 | abid 2341 | . . . 4 ⊢ (x ∈ {x ∣ ψ} ↔ ψ) | |
6 | 4, 5 | imbi12i 316 | . . 3 ⊢ ((x ∈ {x ∣ φ} → x ∈ {x ∣ ψ}) ↔ (φ → ψ)) |
7 | 6 | albii 1566 | . 2 ⊢ (∀x(x ∈ {x ∣ φ} → x ∈ {x ∣ ψ}) ↔ ∀x(φ → ψ)) |
8 | 3, 7 | bitri 240 | 1 ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∈ wcel 1710 {cab 2339 ⊆ wss 3257 |
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-nan 1288 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 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: abss 3335 ssab 3336 ss2abi 3338 ss2abdv 3339 ss2rab 3342 rabss2 3349 |
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