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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 1858 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
4 | spcgf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | notbid 321 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
6 | 1, 3, 5 | spcgf 3538 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓)) |
7 | 6 | con2d 136 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑)) |
8 | df-ex 1782 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
9 | 7, 8 | syl6ibr 255 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 |
This theorem is referenced by: spcegvOLD 3546 rspce 3560 euotd 5368 bnj607 32298 bnj1491 32439 rspcegf 41652 stoweidlem36 42678 stoweidlem46 42688 ichnreuop 43989 ichreuopeq 43990 |
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