|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ss2ab | Structured version Visualization version 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 2906 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
| 2 | nfab1 2906 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
| 3 | 1, 2 | dfssf 3973 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓})) | 
| 4 | abid 2717 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 5 | abid 2717 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
| 6 | 4, 5 | imbi12i 350 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓)) | 
| 7 | 6 | albii 1818 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓)) | 
| 8 | 3, 7 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 ∈ wcel 2107 {cab 2713 ⊆ 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-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-clel 2815 df-nfc 2891 df-ss 3967 | 
| This theorem is referenced by: abss 4062 ssab 4063 ss2rab 4070 rabss2 4077 rabsssn 4667 rabsspr 32521 rabsstp 32522 bj-gabss 36937 clss2lem 43629 ssabf 45110 abssf 45122 cfsetssfset 47073 sprssspr 47473 | 
| Copyright terms: Public domain | W3C validator |