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Mirrors > Home > MPE Home > Th. List > sseq12i | Structured version Visualization version GIF version |
Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
sseq1i.1 | ⊢ 𝐴 = 𝐵 |
sseq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
sseq12i | ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sseq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | sseq12 3661 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ⊆ wss 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-in 3614 df-ss 3621 |
This theorem is referenced by: 3sstr3i 3676 3sstr4i 3677 3sstr3g 3678 3sstr4g 3679 ss2rab 3711 rabsssn 4247 issubgr 26208 pjordi 29160 mdsldmd1i 29318 iuninc 29505 cvmlift2lem12 31422 brtrclfv2 38336 nzss 38833 hoidmvle 41135 ovolval5lem3 41189 fldhmsubc 42409 fldhmsubcALTV 42427 |
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