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Mirrors > Home > MPE Home > Th. List > gencbvex2 | Structured version Visualization version GIF version |
Description: Restatement of gencbvex 3470 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.) |
Ref | Expression |
---|---|
gencbvex2.1 | ⊢ 𝐴 ∈ V |
gencbvex2.2 | ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) |
gencbvex2.3 | ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) |
gencbvex2.4 | ⊢ (𝜃 → ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) |
Ref | Expression |
---|---|
gencbvex2 | ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gencbvex2.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | gencbvex2.2 | . 2 ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | gencbvex2.3 | . 2 ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃)) | |
4 | gencbvex2.4 | . . 3 ⊢ (𝜃 → ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) | |
5 | 3 | biimpac 471 | . . . 4 ⊢ ((𝜒 ∧ 𝐴 = 𝑦) → 𝜃) |
6 | 5 | exlimiv 1889 | . . 3 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝑦) → 𝜃) |
7 | 4, 6 | impbii 201 | . 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦)) |
8 | 1, 2, 3, 7 | gencbvex 3470 | 1 ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 Vcvv 3415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-nf 1747 df-cleq 2771 df-clel 2846 |
This theorem is referenced by: (None) |
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