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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 38893. Version of axc16g 2260 using ax-c11 38888. (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 38932 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥) | |
| 2 | ax-c16 38893 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
| 3 | biidd 262 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑)) | |
| 4 | 3 | dral1-o 38905 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑)) | 
| 5 | 4 | biimprd 248 | . 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 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-11 2157 ax-c5 38884 ax-c4 38885 ax-c7 38886 ax-c10 38887 ax-c11 38888 ax-c9 38891 ax-c16 38893 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 | 
| This theorem is referenced by: ax12inda2 38948 | 
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