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Mirrors > Home > MPE Home > Th. List > ss2abiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ss2abi 3956 as of 28-Jun-2024. (Contributed by NM, 31-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ss2abiOLD.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
ss2abiOLD | ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2ab 3949 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | |
2 | ss2abiOLD.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | mpgbir 1806 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2716 ⊆ wss 3843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 df-in 3850 df-ss 3860 |
This theorem is referenced by: (None) |
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