![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfovd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfov 6936. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nfovd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfovd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐹) |
nfovd.4 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfovd | ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6909 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | nfovd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹) | |
3 | nfovd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfovd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
5 | 3, 4 | nfopd 4641 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
6 | 2, 5 | nffvd 6446 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘〈𝐴, 𝐵〉)) |
7 | 1, 6 | nfcxfrd 2969 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2957 〈cop 4404 ‘cfv 6124 (class class class)co 6906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-iota 6087 df-fv 6132 df-ov 6909 |
This theorem is referenced by: nfov 6936 nfnegd 10597 |
Copyright terms: Public domain | W3C validator |