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

Theorem zfrepclf 5096
Description: An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
Hypotheses
Ref Expression
zfrepclf.1 𝑥𝐴
zfrepclf.2 𝐴 ∈ V
zfrepclf.3 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
Assertion
Ref Expression
zfrepclf 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Distinct variable groups:   𝑦,𝑧,𝐴   𝜑,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem zfrepclf
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 zfrepclf.2 . 2 𝐴 ∈ V
2 zfrepclf.1 . . . . . 6 𝑥𝐴
32nfeq2 2966 . . . . 5 𝑥 𝑣 = 𝐴
4 eleq2 2873 . . . . . 6 (𝑣 = 𝐴 → (𝑥𝑣𝑥𝐴))
5 zfrepclf.3 . . . . . 6 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
64, 5syl6bi 254 . . . . 5 (𝑣 = 𝐴 → (𝑥𝑣 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)))
73, 6alrimi 2180 . . . 4 (𝑣 = 𝐴 → ∀𝑥(𝑥𝑣 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)))
8 nfv 1896 . . . . 5 𝑧𝜑
98axrep5 5094 . . . 4 (∀𝑥(𝑥𝑣 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑣𝜑)))
107, 9syl 17 . . 3 (𝑣 = 𝐴 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑣𝜑)))
114anbi1d 629 . . . . . . 7 (𝑣 = 𝐴 → ((𝑥𝑣𝜑) ↔ (𝑥𝐴𝜑)))
123, 11exbid 2192 . . . . . 6 (𝑣 = 𝐴 → (∃𝑥(𝑥𝑣𝜑) ↔ ∃𝑥(𝑥𝐴𝜑)))
1312bibi2d 344 . . . . 5 (𝑣 = 𝐴 → ((𝑦𝑧 ↔ ∃𝑥(𝑥𝑣𝜑)) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))))
1413albidv 1902 . . . 4 (𝑣 = 𝐴 → (∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑣𝜑)) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))))
1514exbidv 1903 . . 3 (𝑣 = 𝐴 → (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑣𝜑)) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))))
1610, 15mpbid 233 . 2 (𝑣 = 𝐴 → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑)))
171, 16vtocle 3529 1 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1523   = wceq 1525  wex 1765  wcel 2083  wnfc 2935  Vcvv 3440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-rep 5088
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-cleq 2790  df-clel 2865  df-nfc 2937
This theorem is referenced by:  zfrep3cl  5097  zfrep4  5098
  Copyright terms: Public domain W3C validator