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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 4010 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ⊆ wss 3949 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: 3sstr3i 4025 3sstr4i 4026 3sstr3g 4027 3sstr4g 4028 ss2rab 4069 rabsssn 4671 issubgr 28528 pjordi 31426 mdsldmd1i 31584 iuninc 31792 cvmlift2lem12 34305 brtrclfv2 42478 nzss 43076 hoidmvle 45316 fldhmsubc 46982 fldhmsubcALTV 47000 |
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