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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 1855 | . . . 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 3591 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
7 | 6 | con2d 134 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
8 | df-ex 1777 | . 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 1535 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 |
This theorem is referenced by: rspce 3611 euotd 5523 bnj607 34909 bnj1491 35050 rspcegf 44961 stoweidlem36 45992 stoweidlem46 46002 ichnreuop 47397 ichreuopeq 47398 |
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