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Theorem cover2g 33932
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 4602 . . . 4 (𝑏 = 𝐵 𝑏 = 𝐵)
2 cover2g.1 . . . 4 𝐴 = 𝐵
31, 2syl6eqr 2817 . . 3 (𝑏 = 𝐵 𝑏 = 𝐴)
4 rexeq 3287 . . 3 (𝑏 = 𝐵 → (∃𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑦𝐵 (𝑥𝑦𝜑)))
53, 4raleqbidv 3300 . 2 (𝑏 = 𝐵 → (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)))
6 pweq 4318 . . 3 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
73eqeq2d 2775 . . . 4 (𝑏 = 𝐵 → ( 𝑧 = 𝑏 𝑧 = 𝐴))
87anbi1d 623 . . 3 (𝑏 = 𝐵 → (( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
96, 8rexeqbidv 3301 . 2 (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
10 vex 3353 . . 3 𝑏 ∈ V
11 eqid 2765 . . 3 𝑏 = 𝑏
1210, 11cover2 33931 . 2 (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑))
135, 9, 12vtoclbg 3419 1 (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  𝒫 cpw 4315   cuni 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-in 3739  df-ss 3746  df-pw 4317  df-uni 4595
This theorem is referenced by: (None)
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