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| Mirrors > Home > MPE Home > Th. List > eqimssd | Structured version Visualization version GIF version | ||
| Description: Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| eqimssd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| eqimssd | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqimssd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | ssid 4006 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
| 3 | 1, 2 | eqsstrdi 4028 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 | 
| 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 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 | 
| This theorem is referenced by: eqimss 4042 fssrescdmd 7146 f1ocoima 7323 sraassab 21888 evls1maplmhm 22381 fldextrspunlem1 33725 r1peuqusdeg1 35648 selvvvval 42595 stgrnbgr0 47931 | 
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