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| Mirrors > Home > MPE Home > Th. List > nfaba1 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2377. See nfaba1g 2915 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1967, ax-7 2007, ax-12 2177. (Revised by SN, 14-May-2025.) | 
| Ref | Expression | 
|---|---|
| nfaba1 | ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑) | |
| 2 | sbal 2169 | . . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
| 3 | nfa1 2151 | . . . 4 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑 | |
| 4 | 2, 3 | nfxfr 1853 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑 | 
| 5 | 1, 4 | nfxfr 1853 | . 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} | 
| 6 | 5 | nfci 2893 | 1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∀wal 1538 [wsb 2064 ∈ wcel 2108 {cab 2714 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-10 2141 ax-11 2157 | 
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-nfc 2892 | 
| This theorem is referenced by: nfopd 4890 nfimad 6087 nfiota1 6516 nffvd 6918 nfunidALT2 38970 nfunidALT 38971 nfopdALT 38972 setrec1 49210 | 
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