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Theorem zfpow 3955
 Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfpow 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfpow
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 3954 . 2 𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥)
2 elequ1 1616 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
3 elequ1 1616 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3imbi12d 227 . . . . . 6 (𝑤 = 𝑥 → ((𝑤𝑦𝑤𝑧) ↔ (𝑥𝑦𝑥𝑧)))
54cbvalv 1810 . . . . 5 (∀𝑤(𝑤𝑦𝑤𝑧) ↔ ∀𝑥(𝑥𝑦𝑥𝑧))
65imbi1i 231 . . . 4 ((∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ (∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
76albii 1375 . . 3 (∀𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
87exbii 1512 . 2 (∃𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 137 1 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-13 1420  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-pow 3954 This theorem depends on definitions:  df-bi 114  df-nf 1366 This theorem is referenced by:  el  3958
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