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Mirrors > Home > MPE Home > Th. List > 19.9h | Structured version Visualization version GIF version |
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) |
Ref | Expression |
---|---|
19.9h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
19.9h | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9h.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2197 | . 2 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | 19.9 2247 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 ∃wex 1878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-ex 1879 df-nf 1883 |
This theorem is referenced by: bnj1131 31393 bnj1397 31440 |
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