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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 2389. Use the weaker cbvral3vw 3466 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 3202 | . . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒)) |
3 | 2 | cbvralv 3455 | . 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 3467 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
7 | 6 | ralbii 3168 | . 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
8 | 3, 7 | bitri 277 | 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 208 ∀wral 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2896 df-nfc 2966 df-ral 3146 |
This theorem is referenced by: (None) |
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