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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 3176 | . 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 205 ∧ wa 396 ∈ wcel 2106 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-rex 3071 |
This theorem is referenced by: reu3op 6291 opreu2reurex 6293 cnref1o 12968 mndpfo 18647 wspthsnwspthsnon 29167 mdsymlem8 31658 xlt2addrd 31966 elunirnmbfm 33245 satfv0 34344 fmla0xp 34369 icoreelrnab 36230 rrx2xpref1o 47394 |
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