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Mirrors > Home > NFE Home > Th. List > syl6sseq | GIF version |
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
syl6sseq.1 | ⊢ (φ → A ⊆ B) |
syl6sseq.2 | ⊢ B = C |
Ref | Expression |
---|---|
syl6sseq | ⊢ (φ → A ⊆ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6sseq.1 | . 2 ⊢ (φ → A ⊆ B) | |
2 | syl6sseq.2 | . . 3 ⊢ B = C | |
3 | 2 | sseq2i 3297 | . 2 ⊢ (A ⊆ B ↔ A ⊆ C) |
4 | 1, 3 | sylib 188 | 1 ⊢ (φ → A ⊆ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ⊆ wss 3258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 2479 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-ss 3260 |
This theorem is referenced by: syl6sseqr 3319 sspw1 4336 foimacnv 5304 |
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