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Mirrors > Home > MPE Home > Th. List > sbcg | Structured version Visualization version GIF version |
Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3652. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
sbcg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1995 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | sbcgf 3652 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 2145 [wsbc 3588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-v 3353 df-sbc 3589 |
This theorem is referenced by: sbcabel 3667 csbuni 4603 csbxp 5341 sbcfung 6056 fmptsnd 6580 f1od2 29840 bnj89 31128 bnj525 31146 bnj1128 31397 csbwrecsg 33511 csbrdgg 33513 csboprabg 33514 mptsnunlem 33523 topdifinffinlem 33533 relowlpssretop 33550 rdgeqoa 33556 csbfinxpg 33563 sbtru 34241 sbfal 34242 cdlemk40 36727 cdlemkid3N 36743 cdlemkid4 36744 frege70 38754 frege77 38761 frege116 38800 frege118 38802 trsbc 39276 csbxpgOLD 39577 csbxpgVD 39653 csbunigVD 39657 |
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