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Mirrors > Home > MPE Home > Th. List > spesbcd | Structured version Visualization version GIF version |
Description: form of spsbc 3448. (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 3521 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1704 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 |
This theorem is referenced by: euotd 4975 ex-natded9.26 27276 bnj1465 30915 bj-sels 32950 spesbcdi 33925 brtrclfv2 38019 cotrclrcl 38034 |
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