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Mirrors > Home > NFE Home > Th. List > 19.41vv | GIF version |
Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.) |
Ref | Expression |
---|---|
19.41vv | ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.41v 1901 | . . 3 ⊢ (∃y(φ ∧ ψ) ↔ (∃yφ ∧ ψ)) | |
2 | 1 | exbii 1582 | . 2 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(∃yφ ∧ ψ)) |
3 | 19.41v 1901 | . 2 ⊢ (∃x(∃yφ ∧ ψ) ↔ (∃x∃yφ ∧ ψ)) | |
4 | 2, 3 | bitri 240 | 1 ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 |
This theorem is referenced by: 19.41vvv 1903 pm11.07 2115 dfpw12 4301 ltfinex 4464 setconslem4 4734 setconslem6 4736 rabxp 4814 elres 4995 fnov 5591 mpt2mptx 5708 restxp 5786 lecex 6115 |
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