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Mirrors > Home > MPE Home > Th. List > ralrexbid | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) |
Ref | Expression |
---|---|
ralrexbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
ralrexbid | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3142 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | ralrexbid.1 | . . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | |
3 | 2 | imim2i 16 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜃))) |
4 | 3 | pm5.32d 579 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜃))) |
5 | 4 | alexbii 1832 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) |
6 | 1, 5 | sylbi 219 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) |
7 | df-rex 3143 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
8 | df-rex 3143 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃)) | |
9 | 6, 7, 8 | 3bitr4g 316 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-ral 3142 df-rex 3143 |
This theorem is referenced by: dmopab2rex 5779 fiun 7637 f1iun 7638 dmopab3rexdif 32671 |
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