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| Mirrors > Home > MPE Home > Th. List > eqimsscd | Structured version Visualization version GIF version | ||
| Description: Equality implies inclusion, deduction version. (Contributed by SN, 15-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| eqimssd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| eqimsscd | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqimssd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | ssid 4006 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
| 3 | 1, 2 | eqsstrrdi 4029 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | 
| 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: mhplss 22159 precsexlem6 28236 precsexlem7 28237 unitscyglem5 42200 mhphf 42607 isubgrvtxuhgr 47850 | 
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