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Theorem cover2g 37723
Description: Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
Hypothesis
Ref Expression
cover2g.1 𝐴 = 𝐵
Assertion
Ref Expression
cover2g (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
Distinct variable groups:   𝜑,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem cover2g
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 unieq 4918 . . . 4 (𝑏 = 𝐵 𝑏 = 𝐵)
2 cover2g.1 . . . 4 𝐴 = 𝐵
31, 2eqtr4di 2795 . . 3 (𝑏 = 𝐵 𝑏 = 𝐴)
4 rexeq 3322 . . 3 (𝑏 = 𝐵 → (∃𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑦𝐵 (𝑥𝑦𝜑)))
53, 4raleqbidv 3346 . 2 (𝑏 = 𝐵 → (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)))
6 pweq 4614 . . 3 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
73eqeq2d 2748 . . . 4 (𝑏 = 𝐵 → ( 𝑧 = 𝑏 𝑧 = 𝐴))
87anbi1d 631 . . 3 (𝑏 = 𝐵 → (( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
96, 8rexeqbidv 3347 . 2 (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
10 vex 3484 . . 3 𝑏 ∈ V
11 eqid 2737 . . 3 𝑏 = 𝑏
1210, 11cover2 37722 . 2 (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑))
135, 9, 12vtoclbg 3557 1 (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  𝒫 cpw 4600   cuni 4907
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  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-uni 4908
This theorem is referenced by: (None)
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