Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3591 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: 3sstr3i 3606 3sstr4i 3607 3sstr3g 3608 3sstr4g 3609 ss2rab 3641 rabsssn 4162 pjordi 28416 mdsldmd1i 28574 iuninc 28761 cvmlift2lem12 30550 brtrclfv2 37038 nzss 37538 hoidmvle 39490 ovolval5lem3 39544 issubgr 40495 fldhmsubc 41876 fldhmsubcALTV 41895 |
Copyright terms: Public domain | W3C validator |