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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 3996 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⊆ wss 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-in 3945 df-ss 3954 |
This theorem is referenced by: 3sstr3i 4011 3sstr4i 4012 3sstr3g 4013 3sstr4g 4014 ss2rab 4049 rabsssn 4609 issubgr 27055 pjordi 29952 mdsldmd1i 30110 iuninc 30314 cvmlift2lem12 32563 brtrclfv2 40079 nzss 40656 hoidmvle 42889 ovolval5lem3 42943 fldhmsubc 44362 fldhmsubcALTV 44380 |
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