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| 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 | spcimgfi1 3547 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) | 
| 4 | spcimgf.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) | |
| 5 | 3, 4 | mpg 1797 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 | 
| This theorem is referenced by: spcimegf 3551 iooelexlt 37363 | 
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