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Mirrors > Home > MPE Home > Th. List > ss2abiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ss2abi 4024 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 4017 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | |
2 | ss2abiOLD.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | mpgbir 1802 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2710 ⊆ wss 3911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-in 3918 df-ss 3928 |
This theorem is referenced by: (None) |
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