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Mirrors > Home > MPE Home > Th. List > 3sstr3i | Structured version Visualization version GIF version |
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
3sstr3.1 | ⊢ 𝐴 ⊆ 𝐵 |
3sstr3.2 | ⊢ 𝐴 = 𝐶 |
3sstr3.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3i | ⊢ 𝐶 ⊆ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | 3sstr3.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 3979 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | mpbi 229 | 1 ⊢ 𝐶 ⊆ 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3915 |
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 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 |
This theorem is referenced by: ttrclco 9661 cottrcl 9662 odf1o2 19362 leordtval2 22579 uniiccvol 24960 ballotlem2 33128 cotrcltrcl 42071 |
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