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Mirrors > Home > MPE Home > Th. List > cbvral3v | Structured version Visualization version GIF version |
Description: Change bound variables of triple restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbvral3vw 3239 when possible. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvral3v.1 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) |
cbvral3v.2 | ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) |
cbvral3v.3 | ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvral3v | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvral3v.1 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | |
2 | 1 | 2ralbidv 3217 | . . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒)) |
3 | 2 | cbvralv 3359 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒) |
4 | cbvral3v.2 | . . . 4 ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) | |
5 | cbvral3v.3 | . . . 4 ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) | |
6 | 4, 5 | cbvral2v 3363 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
7 | 6 | ralbii 3092 | . 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
8 | 3, 7 | bitri 274 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wral 3060 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clel 2809 df-nfc 2884 df-ral 3061 |
This theorem is referenced by: (None) |
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