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Mirrors > Home > MPE Home > Th. List > cbv2h | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.) |
Ref | Expression |
---|---|
cbv2h.1 | ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
cbv2h.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
cbv2h.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbv2h | ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv2h.1 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) | |
2 | cbv2h.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | cbv2h.3 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
4 | biimp 207 | . . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 1, 2, 5 | cbv1h 2420 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) |
7 | equcomi 2121 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
8 | biimpr 212 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
9 | 7, 3, 8 | syl56 36 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓))) |
10 | 2, 1, 9 | cbv1h 2420 | . . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
11 | 10 | alcoms 2208 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓)) |
12 | 6, 11 | impbid 204 | 1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ex 1879 df-nf 1883 |
This theorem is referenced by: cbv2 2422 eujustALT 2643 |
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