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Mirrors > Home > MPE Home > Th. List > nfor | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nf.1 | ⊢ Ⅎ𝑥𝜑 |
nf.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
nfor | ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-or 384 | . 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
2 | nf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfn 1768 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
4 | nf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfim 1813 | . 2 ⊢ Ⅎ𝑥(¬ 𝜑 → 𝜓) |
6 | 1, 5 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-or 384 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nf3or 1823 axi12 2588 nfun 3731 nfpr 4179 rabsnifsb 4201 disjxun 4581 fsuppmapnn0fiubex 12654 nfsum1 14268 nfsum 14269 nfcprod1 14479 nfcprod 14480 fdc1 32712 dvdsrabdioph 36392 disjinfi 38375 iundjiun 39353 |
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