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Mirrors > Home > MPE Home > Th. List > Mathboxes > e2ebind | Structured version Visualization version GIF version |
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 40890 is derived from e2ebindVD 41239. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
e2ebind | ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2150 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | 19.9 2201 | . . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) |
3 | biidd 264 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
4 | 3 | drex1 2459 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑)) |
5 | 4 | drex2 2460 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)) |
6 | excom 2165 | . . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
7 | 5, 6 | syl6bb 289 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)) |
8 | 2, 7 | syl5rbbr 288 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
9 | 8 | aecoms 2446 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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