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Mirrors > Home > ILE Home > Th. List > hbsb3 | GIF version |
Description: If y is not free in φ, x is not free in [y / x]φ. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbsb3.1 | ⊢ (φ → ∀yφ) |
Ref | Expression |
---|---|
hbsb3 | ⊢ ([y / x]φ → ∀x[y / x]φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb3.1 | . . 3 ⊢ (φ → ∀yφ) | |
2 | 1 | sbimi 1644 | . 2 ⊢ ([y / x]φ → [y / x]∀yφ) |
3 | hbsb2a 1684 | . 2 ⊢ ([y / x]∀yφ → ∀x[y / x]φ) | |
4 | 2, 3 | syl 14 | 1 ⊢ ([y / x]φ → ∀x[y / x]φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-11 1394 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: nfs1 1687 sbcof2 1688 ax16 1691 sb8h 1731 sb8eh 1732 ax16ALT 1736 |
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