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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 3501 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
4 | spcimgf.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
5 | 3, 4 | mpg 1896 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1654 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 Ⅎwnfc 2956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 |
This theorem is referenced by: spcimegf 3504 iooelexlt 33754 |
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