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Mirrors > Home > MPE Home > Th. List > ss2abdvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ss2abdv 4052 as of 28-Jun-2024. (Contributed by NM, 29-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ss2abdvOLD.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2abdvOLD | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2abdvOLD.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimiv 1922 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | ss2ab 4048 | . 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒)) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 {cab 2701 ⊆ wss 3940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-v 3468 df-in 3947 df-ss 3957 |
This theorem is referenced by: (None) |
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