![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbceq1a | Structured version Visualization version GIF version |
Description: Equality theorem for class substitution. Class version of sbequ12 2249. (Contributed by NM, 26-Sep-2003.) |
Ref | Expression |
---|---|
sbceq1a | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbid 2253 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
2 | dfsbcq2 3794 | . 2 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 1, 2 | bitr3id 285 | 1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 [wsb 2062 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbceq2a 3803 elrabsf 3840 cbvralcsf 3953 reusngf 4679 rexreusng 4684 reuprg0 4707 rmosn 4724 rabsnifsb 4727 euotd 5523 reuop 6315 frpoinsg 6366 wfisgOLD 6377 elfvmptrab1w 7043 elfvmptrab1 7044 ralrnmpt 7116 riotass2 7418 riotass 7419 oprabv 7493 elovmporab 7679 elovmporab1w 7680 elovmporab1 7681 ovmpt3rabdm 7692 elovmpt3rab1 7693 tfisg 7875 tfindes 7884 sbcopeq1a 8073 sbcoteq1a 8075 mpoxopoveq 8243 findcard2 9203 ac6sfi 9318 indexfi 9398 frinsg 9789 nn0ind-raph 12716 fzrevral 13649 wrdind 14757 wrd2ind 14758 prmind2 16719 elmptrab 23851 isfildlem 23881 2sqreulem4 27513 gropd 29063 grstructd 29064 rspc2daf 32495 opreu2reuALT 32505 ifeqeqx 32563 wrdt2ind 32923 bnj919 34760 bnj976 34770 bnj1468 34839 bnj110 34851 bnj150 34869 bnj151 34870 bnj607 34909 bnj873 34917 bnj849 34918 bnj1388 35026 setinds 35760 dfon2lem1 35765 rdgssun 37361 indexdom 37721 sdclem2 37729 sdclem1 37730 fdc1 37733 riotasv2s 38940 elimhyps 38943 sbccomieg 42781 rexrabdioph 42782 rexfrabdioph 42783 aomclem6 43048 pm13.13a 44403 pm13.13b 44404 pm13.14 44405 tratrb 44534 uzwo4 44993 or2expropbilem2 46983 reuf1odnf 47057 ich2exprop 47396 ichnreuop 47397 ichreuopeq 47398 prproropreud 47434 reupr 47447 reuopreuprim 47451 |
Copyright terms: Public domain | W3C validator |