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| 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 1856 | . . . 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 3590 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) | 
| 7 | 6 | con2d 134 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) | 
| 8 | df-ex 1779 | . 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 ↔ wb 206 ∀wal 1537 = wceq 1539 ∃wex 1778 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-cleq 2728 df-clel 2815 df-nfc 2891 | 
| This theorem is referenced by: rspce 3610 euotd 5517 bnj607 34931 bnj1491 35072 rspcegf 45033 stoweidlem36 46056 stoweidlem46 46066 ichnreuop 47464 ichreuopeq 47465 | 
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