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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 42937 is derived from e2ebindVD 43286. (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 | biidd 262 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
2 | 1 | drex1 2440 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑)) |
3 | 2 | drex2 2441 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)) |
4 | excom 2163 | . . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
5 | 3, 4 | bitrdi 287 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)) |
6 | nfe1 2148 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
7 | 6 | 19.9 2199 | . . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) |
8 | 5, 7 | bitr3di 286 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
9 | 8 | aecoms 2427 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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