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Theorem cbvexfo 7283
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
Hypothesis
Ref Expression
cbvfo.1 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexfo (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐵(𝑥)

Proof of Theorem cbvexfo
StepHypRef Expression
1 cbvfo.1 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
21notbid 318 . . . 4 ((𝐹𝑥) = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32cbvfo 7282 . . 3 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓))
43notbid 318 . 2 (𝐹:𝐴onto𝐵 → (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓))
5 dfrex2 3067 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
6 dfrex2 3067 . 2 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
74, 5, 63bitr4g 314 1 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wral 3055  wrex 3064  ontowfo 6534  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544
This theorem is referenced by:  f1oweALT  7955  deg1ldg  25979
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