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Mirrors > Home > MPE Home > Th. List > gencbval | Structured version Visualization version GIF version |
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) |
Ref | Expression |
---|---|
gencbval.1 | ⊢ 𝐴 ∈ V |
gencbval.2 | ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) |
gencbval.3 | ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) |
gencbval.4 | ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) |
Ref | Expression |
---|---|
gencbval | ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gencbval.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | gencbval.2 | . . . . 5 ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 321 | . . . 4 ⊢ (𝐴 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | gencbval.3 | . . . 4 ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) | |
5 | gencbval.4 | . . . 4 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) | |
6 | 1, 3, 4, 5 | gencbvex 3453 | . . 3 ⊢ (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓)) |
7 | exanali 1866 | . . 3 ⊢ (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒 → 𝜑)) | |
8 | exanali 1866 | . . 3 ⊢ (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃 → 𝜓)) | |
9 | 6, 7, 8 | 3bitr3i 304 | . 2 ⊢ (¬ ∀𝑥(𝜒 → 𝜑) ↔ ¬ ∀𝑦(𝜃 → 𝜓)) |
10 | 9 | con4bii 324 | 1 ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 = wceq 1542 ∃wex 1786 ∈ wcel 2114 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-nf 1791 df-cleq 2730 df-clel 2811 |
This theorem is referenced by: (None) |
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