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Mirrors > Home > ILE Home > Th. List > nfsbd | GIF version |
Description: Deduction version of nfsb 1958. (Contributed by NM, 15-Feb-2013.) |
Ref | Expression |
---|---|
nfsbd.1 | ⊢ Ⅎ𝑥𝜑 |
nfsbd.2 | ⊢ (𝜑 → Ⅎ𝑧𝜓) |
Ref | Expression |
---|---|
nfsbd | ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsbd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1530 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfsbd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
4 | 3 | alimi 1466 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓) |
5 | nfsbt 1988 | . 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) | |
6 | 2, 4, 5 | 3syl 17 | 1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 Ⅎwnf 1471 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: nfeud 2054 nfabd 2352 nfraldya 2525 nfrexdya 2526 cbvrald 15018 |
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