Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvexfo Structured version   Visualization version   GIF version

Theorem cbvexfo 6800
 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 310 . . . 4 ((𝐹𝑥) = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32cbvfo 6799 . . 3 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ 𝜓))
43notbid 310 . 2 (𝐹:𝐴onto𝐵 → (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓))
5 dfrex2 3204 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
6 dfrex2 3204 . 2 (∃𝑦𝐵 𝜓 ↔ ¬ ∀𝑦𝐵 ¬ 𝜓)
74, 5, 63bitr4g 306 1 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1658  ∀wral 3117  ∃wrex 3118  –onto→wfo 6121  ‘cfv 6123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fo 6129  df-fv 6131 This theorem is referenced by:  f1oweALT  7412  deg1ldg  24251
 Copyright terms: Public domain W3C validator