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| Mirrors > Home > MPE Home > Th. List > 2rexbiia | Structured version Visualization version GIF version | ||
| Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.) |
| Ref | Expression |
|---|---|
| 2rexbiia.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| 2rexbiia | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2rexbiia.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | rexbidva 3177 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) |
| 3 | 2 | rexbiia 3092 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-rex 3071 |
| This theorem is referenced by: reu3op 6312 opreu2reurex 6314 cnref1o 13027 mndpfo 18770 wspthsnwspthsnon 29936 mdsymlem8 32429 xlt2addrd 32762 elunirnmbfm 34253 satfv0 35363 fmla0xp 35388 icoreelrnab 37355 fimgmcyclem 42543 rrx2xpref1o 48639 |
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