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Mirrors > Home > MPE Home > Th. List > nfovd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfov 7189. (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 7162 | . 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 4823 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
6 | 2, 5 | nffvd 6685 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘〈𝐴, 𝐵〉)) |
7 | 1, 6 | nfcxfrd 2979 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2964 〈cop 4576 ‘cfv 6358 (class class class)co 7159 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: nfov 7189 nfnegd 10884 |
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