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Mirrors > Home > MPE Home > Th. List > axc16gb | Structured version Visualization version GIF version |
Description: Biconditional strengthening of axc16g 2261. (Contributed by NM, 15-May-1993.) |
Ref | Expression |
---|---|
axc16gb | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc16g 2261 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
2 | sp 2182 | . 2 ⊢ (∀𝑧𝜑 → 𝜑) | |
3 | 1, 2 | impbid1 227 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: sbalOLD 2575 |
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