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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 1797 | . 2 ⊢ ∀𝑥𝜑 |
3 | spsbc 3733 | . 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 1536 ∈ wcel 2111 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: iota4an 6306 tfinds2 7558 wunnat 17218 catcfuccl 17361 dprdval 19118 opsbc2ie 30246 bj-sbceqgALT 34343 f1omptsnlem 34753 mptsnunlem 34755 topdifinffinlem 34764 relowlpssretop 34781 cdlemk35s 38233 cdlemk39s 38235 cdlemk42 38237 frege92 40656 |
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