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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexcom4f | Structured version Visualization version GIF version |
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
ralcom4f.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
rexcom4f | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom4f.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2901 | . . 3 ⊢ Ⅎ𝑥V | |
3 | 1, 2 | rexcomf 3299 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑) |
4 | rexv 3506 | . . 3 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑) | |
5 | 4 | rexbii 3090 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦𝜑) |
6 | rexv 3506 | . 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | |
7 | 3, 5, 6 | 3bitr3i 301 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1774 Ⅎwnfc 2886 ∃wrex 3066 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-11 2153 ax-12 2173 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1538 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-v 3479 |
This theorem is referenced by: (None) |
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