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| Mirrors > Home > MPE Home > Th. List > dfssf | Structured version Visualization version GIF version | ||
| Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) Avoid ax-13 2376. (Revised by GG, 19-May-2023.) | 
| Ref | Expression | 
|---|---|
| dfssf.1 | ⊢ Ⅎ𝑥𝐴 | 
| dfssf.2 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| dfssf | ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ss 3967 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) | |
| 2 | dfssf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | nfcri 2896 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | 
| 4 | dfssf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 5 | 4 | nfcri 2896 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 | 
| 6 | 3, 5 | nfim 1895 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) | 
| 7 | nfv 1913 | . . 3 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) | |
| 8 | eleq1w 2823 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
| 9 | eleq1w 2823 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
| 10 | 8, 9 | imbi12d 344 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))) | 
| 11 | 6, 7, 10 | cbvalv1 2342 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| 12 | 1, 11 | bitri 275 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 ∈ wcel 2107 Ⅎwnfc 2889 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-clel 2815 df-nfc 2891 df-ss 3967 | 
| This theorem is referenced by: dfss3f 3974 ssrd 3987 ssrmof 4050 ss2ab 4061 rankval4 9908 rabexgfGS 32519 ballotth 34541 dvcosre 45932 itgsinexplem1 45974 | 
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