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Mirrors > Home > MPE Home > Th. List > ralcom4OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralcom4 3235 as of 26-Aug-2023. Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralcom4OLD | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3354 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑) | |
2 | ralv 3519 | . . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑) | |
3 | 2 | ralbii 3165 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑) |
4 | ralv 3519 | . 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | |
5 | 1, 3, 4 | 3bitr3i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 ∀wral 3138 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-11 2161 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-v 3496 |
This theorem is referenced by: (None) |
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