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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc16g-o | Structured version Visualization version GIF version |
Description: A generalization of Axiom ax-c16 37757. Version of axc16g 2251 using ax-c11 37752. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc16g-o | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev-o 37796 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥) | |
2 | ax-c16 37757 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
3 | biidd 261 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑)) | |
4 | 3 | dral1-o 37769 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑)) |
5 | 4 | biimprd 247 | . 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑)) |
6 | 1, 2, 5 | sylsyld 61 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-11 2154 ax-c5 37748 ax-c4 37749 ax-c7 37750 ax-c10 37751 ax-c11 37752 ax-c9 37755 ax-c16 37757 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 |
This theorem is referenced by: ax12inda2 37812 |
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