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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 1857 | . . . 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 3550 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
| 7 | 6 | con2d 134 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
| 8 | df-ex 1780 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
| 9 | 7, 8 | imbitrrdi 252 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 |
| This theorem is referenced by: (None) |
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