Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hbsb | 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. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker hbsbw 2173 when possible. (Contributed by NM, 12-Aug-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbsb.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
hbsb | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb.1 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | nf5i 2146 | . . 3 ⊢ Ⅎ𝑧𝜑 |
3 | 2 | nfsb 2526 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
4 | 3 | nf5ri 2193 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: hbabg 2726 hblemg 2868 |
Copyright terms: Public domain | W3C validator |