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Mirrors > Home > MPE Home > Th. List > spesbcd | Structured version Visualization version GIF version |
Description: form of spsbc 3694. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Ref | Expression |
---|---|
spesbcd.1 | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) |
Ref | Expression |
---|---|
spesbcd | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spesbcd.1 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | |
2 | spesbc 3768 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1742 [wsbc 3681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-v 3417 df-sbc 3682 |
This theorem is referenced by: euotd 5259 ex-natded9.26 27976 bnj1465 31770 spesbcdi 34848 brtrclfv2 39441 cotrclrcl 39456 |
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