![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > spcegf | Structured version Visualization version GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.) |
Ref | Expression |
---|---|
spcgf.1 | ⊢ Ⅎ𝑥𝐴 |
spcgf.2 | ⊢ Ⅎ𝑥𝜓 |
spcgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcegf | ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | spcgf.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | nfn 1852 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
4 | spcgf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | notbid 318 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
6 | 1, 3, 5 | spcgf 3573 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
7 | 6 | con2d 134 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
8 | df-ex 1774 | . 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 ↔ wb 205 ∀wal 1531 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-v 3468 |
This theorem is referenced by: rspce 3593 euotd 5504 bnj607 34446 bnj1491 34587 rspcegf 44257 stoweidlem36 45298 stoweidlem46 45308 ichnreuop 46686 ichreuopeq 46687 |
Copyright terms: Public domain | W3C validator |