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Mirrors > Home > MPE Home > Th. List > spcimegf | Structured version Visualization version GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimgf.1 | ⊢ Ⅎ𝑥𝐴 |
spcimgf.2 | ⊢ Ⅎ𝑥𝜓 |
spcimegf.3 | ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) |
Ref | Expression |
---|---|
spcimegf | ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | spcimgf.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | nfn 1848 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
4 | spcimegf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
5 | 4 | con3d 155 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓)) |
6 | 1, 3, 5 | spcimgf 3585 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
7 | 6 | con2d 136 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
8 | df-ex 1772 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
9 | 7, 8 | syl6ibr 253 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 |
This theorem is referenced by: (None) |
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