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Mirrors > Home > MPE Home > Th. List > sbcth | Structured version Visualization version GIF version |
Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.) |
Ref | Expression |
---|---|
sbcth.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
sbcth | ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcth.1 | . . 3 ⊢ 𝜑 | |
2 | 1 | ax-gen 1796 | . 2 ⊢ ∀𝑥𝜑 |
3 | spsbc 3787 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | |
4 | 2, 3 | mpi 20 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∈ wcel 2114 [wsbc 3774 |
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-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-sbc 3775 |
This theorem is referenced by: iota4an 6339 tfinds2 7580 wunnat 17228 catcfuccl 17371 dprdval 19127 opsbc2ie 30241 bj-sbceqgALT 34221 f1omptsnlem 34619 mptsnunlem 34621 topdifinffinlem 34630 relowlpssretop 34647 cdlemk35s 38075 cdlemk39s 38077 cdlemk42 38079 frege92 40308 |
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