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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 1896 | . 2 ⊢ ∀𝑥𝜑 |
3 | spsbc 3674 | . 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 1656 ∈ wcel 2166 [wsbc 3661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-12 2222 ax-13 2390 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1662 df-ex 1881 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-v 3415 df-sbc 3662 |
This theorem is referenced by: iota4an 6104 tfinds2 7323 wunnat 16967 catcfuccl 17110 dprdval 18755 bj-sbceqgALT 33417 f1omptsnlem 33728 mptsnunlem 33730 topdifinffinlem 33739 relowlpssretop 33756 cnfinltrel 33785 cdlemk35s 37011 cdlemk39s 37013 cdlemk42 37015 frege92 39088 |
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