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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 2982 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | nfab1 2982 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
3 | 1, 2 | dfss2f 3961 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓})) |
4 | abid 2806 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | abid 2806 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
6 | 4, 5 | imbi12i 353 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓)) |
7 | 6 | albii 1819 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓)) |
8 | 3, 7 | bitri 277 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 ∈ wcel 2113 {cab 2802 ⊆ wss 3939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-in 3946 df-ss 3955 |
This theorem is referenced by: abss 4043 ssab 4044 ss2abi 4046 ss2abdv 4047 ss2rab 4050 rabss2 4057 rabsssn 4610 clss2lem 39977 ssabf 41372 abssf 41384 sprssspr 43650 |
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