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Mirrors > Home > MPE Home > Th. List > 3sstr4g | Structured version Visualization version 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 3594 | . 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵) |
5 | 1, 4 | sylibr 223 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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: rabss2 3648 unss2 3746 sslin 3801 intss 4433 ssopab2 4926 xpss12 5148 coss1 5199 coss2 5200 cnvss 5216 cnvssOLD 5217 rnss 5275 ssres 5344 ssres2 5345 imass1 5419 imass2 5420 predpredss 5603 ssoprab2 6609 ressuppss 7201 tposss 7240 onovuni 7326 ss2ixp 7807 fodomfi 8124 coss12d 13559 isumsplit 14411 isumrpcl 14414 cvgrat 14454 gsumzf1o 18136 gsumzmhm 18160 gsumzinv 18168 dsmmsubg 19906 qustgpopn 21733 metnrmlem2 22471 ovolsslem 23059 uniioombllem3 23159 ulmres 23946 xrlimcnp 24495 pntlemq 25090 sspba 26966 shlej2i 27622 chpssati 28606 mptssALT 28857 bnj1408 30358 subfacp1lem6 30421 mthmpps 30733 aomclem4 36645 cotrclrcl 37053 cusgredg 40646 fldc 41875 fldcALTV 41894 |
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