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Mirrors > Home > MPE Home > Th. List > 19.9 | Structured version Visualization version GIF version |
Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2202 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1988 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) |
Ref | Expression |
---|---|
19.9.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.9 | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 19.9t 2204 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 |
This theorem is referenced by: exlimd 2218 19.19 2231 19.36 2232 19.41 2237 19.44 2239 19.45 2240 19.9h 2294 dfid3 5462 fsplitOLD 7813 bnj1189 32281 bj-exexbiex 34034 bj-exalbial 34036 ax6e2ndeq 40913 e2ebind 40917 ax6e2ndeqVD 41263 e2ebindVD 41266 e2ebindALT 41283 ax6e2ndeqALT 41285 |
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