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Mirrors > Home > NFE Home > Th. List > 3sstr4g | GIF version |
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
3sstr4g.1 | ⊢ (φ → A ⊆ B) |
3sstr4g.2 | ⊢ C = A |
3sstr4g.3 | ⊢ D = B |
Ref | Expression |
---|---|
3sstr4g | ⊢ (φ → C ⊆ D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr4g.1 | . 2 ⊢ (φ → A ⊆ B) | |
2 | 3sstr4g.2 | . . 3 ⊢ C = A | |
3 | 3sstr4g.3 | . . 3 ⊢ D = B | |
4 | 2, 3 | sseq12i 3297 | . 2 ⊢ (C ⊆ D ↔ A ⊆ B) |
5 | 1, 4 | sylibr 203 | 1 ⊢ (φ → C ⊆ D) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ⊆ wss 3257 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: rabss2 3349 unss2 3434 sslin 3481 pw1ss 4169 ssopab2 4712 xpss12 4855 coss1 4872 coss2 4873 cnvss 4885 rnss 4959 ssres 4990 ssres2 4991 imass1 5023 imass2 5024 mapss 6027 sbthlem1 6203 |
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