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Mirrors > Home > ILE Home > Th. List > sseqtrd | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
sseqtrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sseqtrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sseqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | sseq2d 3200 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)) |
4 | 1, 3 | mpbid 147 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ⊆ wss 3144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 |
This theorem is referenced by: sseqtrrd 3209 fssdmd 5398 resasplitss 5414 nnaword2 6539 erssxp 6582 phpm 6893 nnnninfeq 7156 ioodisj 10023 subsubm 12935 subsubg 13136 trivsubgd 13139 trivnsgd 13156 subsubrng 13561 subrgugrp 13587 subsubrg 13592 islssmd 13675 lspun 13718 lspssp 13719 lsslsp 13745 tgcl 14024 basgen 14040 bastop1 14043 bastop2 14044 clsss2 14089 topssnei 14122 cnntr 14185 txbasval 14227 neitx 14228 cnmpt1res 14256 cnmpt2res 14257 imasnopn 14259 hmeontr 14273 tgioo 14506 reldvg 14608 dvfvalap 14610 dvbss 14614 dvcnp2cntop 14623 dvaddxxbr 14625 dvmulxxbr 14626 dvcj 14633 |
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