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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 1795 | . 2 ⊢ ∀𝑥𝜑 |
| 3 | spsbc 3801 | . 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 1538 ∈ wcel 2108 [wsbc 3788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: iota4an 6543 tfinds2 7885 wunnat 18004 catcfuccl 18163 dprdval 20023 opsbc2ie 32495 bj-sbceqgALT 36903 f1omptsnlem 37337 mptsnunlem 37339 topdifinffinlem 37348 relowlpssretop 37365 cdlemk35s 40939 cdlemk39s 40941 cdlemk42 40943 frege92 43968 |
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