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Mirrors > Home > MPE Home > Th. List > nfsb | Structured version Visualization version GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfsb.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc16nf 2122 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) | |
2 | nfsb.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | 2 | nfsb4 2378 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
4 | 1, 3 | pm2.61i 175 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1473 Ⅎwnf 1699 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: hbsb 2429 sb10f 2444 2sb8e 2455 sb8eu 2491 2mo 2539 cbvralf 3141 cbvreu 3145 cbvralsv 3158 cbvrexsv 3159 cbvrab 3171 cbvreucsf 3533 cbvrabcsf 3534 cbvopab1 4655 cbvmptf 4676 cbvmpt 4677 ralxpf 5190 cbviota 5773 sb8iota 5775 cbvriota 6521 dfoprab4f 7117 mo5f 28708 ax11-pm2 32011 2sb5nd 37797 |
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