Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > spesbcdi | Structured version Visualization version GIF version |
Description: A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
Ref | Expression |
---|---|
spesbcdi.1 | ⊢ (𝜑 → 𝜓) |
spesbcdi.2 | ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) |
Ref | Expression |
---|---|
spesbcdi | ⊢ (𝜑 → ∃𝑥𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spesbcdi.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | spesbcdi.2 | . . 3 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) | |
3 | 1, 2 | sylibr 224 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒) |
4 | 3 | spesbcd 3522 | 1 ⊢ (𝜑 → ∃𝑥𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∃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: (None) |
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