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Mirrors > Home > MPE Home > Th. List > nfsbc | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker nfsbcw 3707 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsbc.1 | ⊢ Ⅎ𝑥𝐴 |
nfsbc.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfsbc | ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1811 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfsbc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfsbc.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfsbcd 3709 | . 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑) |
7 | 6 | mptru 1549 | 1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1543 Ⅎwnf 1790 Ⅎwnfc 2880 [wsbc 3685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2373 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-sbc 3686 |
This theorem is referenced by: cbvralcsf 3842 ralrnmpt 6885 elovmporab1 7422 opreu2reuALT 30412 |
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