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Theorem cover2g 36224
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 4880 . . . 4 (𝑏 = 𝐵 𝑏 = 𝐵)
2 cover2g.1 . . . 4 𝐴 = 𝐵
31, 2eqtr4di 2791 . . 3 (𝑏 = 𝐵 𝑏 = 𝐴)
4 rexeq 3309 . . 3 (𝑏 = 𝐵 → (∃𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑦𝐵 (𝑥𝑦𝜑)))
53, 4raleqbidv 3318 . 2 (𝑏 = 𝐵 → (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)))
6 pweq 4578 . . 3 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
73eqeq2d 2744 . . . 4 (𝑏 = 𝐵 → ( 𝑧 = 𝑏 𝑧 = 𝐴))
87anbi1d 631 . . 3 (𝑏 = 𝐵 → (( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
96, 8rexeqbidv 3319 . 2 (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
10 vex 3451 . . 3 𝑏 ∈ V
11 eqid 2733 . . 3 𝑏 = 𝑏
1210, 11cover2 36223 . 2 (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑))
135, 9, 12vtoclbg 3530 1 (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  𝒫 cpw 4564   cuni 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870
This theorem is referenced by: (None)
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