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Mirrors > Home > ILE Home > Th. List > 2exbii | GIF version |
Description: Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |
Ref | Expression |
---|---|
exbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2exbii | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | exbii 1569 | . 2 ⊢ (∃𝑦𝜑 ↔ ∃𝑦𝜓) |
3 | 2 | exbii 1569 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 3exbii 1571 19.42vvvv 1867 3exdistr 1869 cbvex4v 1882 ee4anv 1886 ee8anv 1887 sbel2x 1951 2eu4 2070 rexcomf 2570 reean 2576 ceqsex3v 2702 ceqsex4v 2703 ceqsex8v 2705 copsexg 4136 opelopabsbALT 4151 opabm 4172 uniuni 4342 rabxp 4546 elxp3 4563 elvv 4571 elvvv 4572 rexiunxp 4651 elcnv2 4687 cnvuni 4695 coass 5027 fununi 5161 dfmpt3 5215 dfoprab2 5786 dmoprab 5820 rnoprab 5822 mpomptx 5830 resoprab 5835 ovi3 5875 ov6g 5876 oprabex3 5995 xpassen 6692 enq0enq 7207 enq0sym 7208 enq0tr 7210 ltresr 7615 axaddf 7644 axmulf 7645 |
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