Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3sstr3g | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.) |
Ref | Expression |
---|---|
3sstr3g.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3sstr3g.2 | ⊢ 𝐴 = 𝐶 |
3sstr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3g | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3g.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 3sstr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3g.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 3997 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | sylib 220 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 |
This theorem is referenced by: complss 4123 uniintsn 4913 fpwwe2lem13 10064 hmeocls 22376 hmeontr 22377 usgrumgruspgr 26965 chsscon3i 29238 pjss1coi 29940 mdslmd2i 30107 satffunlem2lem2 32653 ssbnd 35081 bnd2lem 35084 trclubgNEW 39998 nzss 40669 |
Copyright terms: Public domain | W3C validator |