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Mirrors > Home > MPE Home > Th. List > r19.26-3 | Structured version Visualization version GIF version |
Description: Version of r19.26 3112 with three quantifiers. (Contributed by FL, 22-Nov-2010.) |
Ref | Expression |
---|---|
r19.26-3 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1090 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
2 | 1 | ralbii 3094 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
3 | r19.26 3112 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒)) | |
4 | r19.26 3112 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 4 | anbi1i 625 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒)) |
6 | df-3an 1090 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒)) | |
7 | 5, 6 | bitr4i 278 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒)) |
8 | 2, 3, 7 | 3bitri 297 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-ral 3063 |
This theorem is referenced by: sgrp2rid2ex 18808 axeuclid 28221 axcontlem8 28229 stoweidlem60 44776 |
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