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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 7388. (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 7361 | . 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 4848 | . . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩) |
6 | 2, 5 | nffvd 6855 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩)) |
7 | 1, 6 | nfcxfrd 2907 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnfc 2888 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: nfov 7388 nfnegd 11397 |
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