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Mirrors > Home > MPE Home > Th. List > nfsbcd | Structured version Visualization version GIF version |
Description: Deduction version of nfsbc 3816. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfsbcdw 3812 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsbcd.1 | ⊢ Ⅎ𝑦𝜑 |
nfsbcd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfsbcd.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfsbcd | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3792 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
2 | nfsbcd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfsbcd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfsbcd.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabd 2926 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 2, 5 | nfeld 2915 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfxfrd 1851 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 [wsbc 3791 |
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-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-sbc 3792 |
This theorem is referenced by: nfsbc 3816 nfcsbd 3934 sbcnestgf 4432 |
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