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Mirrors > Home > MPE Home > Th. List > rexbi | Structured version Visualization version GIF version |
Description: Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.) |
Ref | Expression |
---|---|
rexbi | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 214 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | ralimi 3087 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) |
3 | rexim 3172 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
5 | biimpr 219 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
6 | 5 | ralimi 3087 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) |
7 | rexim 3172 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜑)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜑)) |
9 | 4, 8 | impbid 211 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wral 3064 ∃wrex 3065 |
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 397 df-ex 1783 df-ral 3069 df-rex 3070 |
This theorem is referenced by: ralrexbid 3255 |
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