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Mirrors > Home > MPE Home > Th. List > ralrexbidOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralrexbid 3103 as of 4-Nov-2024. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralrexbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
ralrexbidOLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3059 | . . 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 575 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜃))) |
5 | 4 | alexbii 1827 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) |
6 | 1, 5 | sylbi 216 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) |
7 | df-rex 3068 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
8 | df-rex 3068 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃)) | |
9 | 6, 7, 8 | 3bitr4g 313 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-ral 3059 df-rex 3068 |
This theorem is referenced by: (None) |
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