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Mirrors > Home > ILE Home > Th. List > ax6blem | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1541 compared to hbnt 1543. (Contributed by GD, 27-Jan-2018.) |
Ref | Expression |
---|---|
ax6blem.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
ax6blem | ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6blem.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | id 19 | . . . 4 ⊢ (𝜑 → 𝜑) | |
3 | 1, 2 | exlimih 1484 | . . 3 ⊢ (∃𝑥𝜑 → 𝜑) |
4 | 3 | con3i 562 | . 2 ⊢ (¬ 𝜑 → ¬ ∃𝑥𝜑) |
5 | alnex 1388 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | 4, 5 | sylibr 137 | 1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 ∃wex 1381 |
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-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie2 1383 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 |
This theorem is referenced by: ax6b 1541 |
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