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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 1853 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
4 | spcimegf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
5 | 4 | con3d 152 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓)) |
6 | 1, 3, 5 | spcimgf 3574 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
7 | 6 | con2d 134 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
8 | df-ex 1775 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
9 | 7, 8 | imbitrrdi 251 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 = wceq 1534 ∃wex 1774 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-v 3471 |
This theorem is referenced by: (None) |
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