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Mirrors > Home > MPE Home > Th. List > spcimgf | Structured version Visualization version GIF version |
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimgf.1 | ⊢ Ⅎ𝑥𝐴 |
spcimgf.2 | ⊢ Ⅎ𝑥𝜓 |
spcimgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spcimgf | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | spcimgf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | spcimgft 3583 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
4 | spcimgf.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1789 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 |
This theorem is referenced by: spcimegf 3586 iooelexlt 34525 |
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