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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbyfrbi | Structured version Visualization version GIF version |
Description: Version of bj-hbxfrbi 33963 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.) |
Ref | Expression |
---|---|
bj-hbyfrbi | ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbi 1847 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) | |
2 | 1 | adantl 484 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) |
3 | simpl 485 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | imbi12d 347 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: bj-nnfbi 34057 |
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