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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12w | Structured version Visualization version GIF version |
Description: The general statement that ax12w 1997 proves. (Contributed by BJ, 20-Mar-2020.) |
Ref | Expression |
---|---|
bj-ax12w.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bj-ax12w.2 | ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
bj-ax12w | ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ax12w.2 | . . 3 ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) | |
2 | 1 | spw 1954 | . 2 ⊢ (∀𝑦𝜓 → 𝜓) |
3 | bj-ax12w.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | bj-ax12wlem 31807 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) |
5 | 2, 4 | syl5 33 | 1 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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