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Theorem axrep2 4701
 Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.)
Assertion
Ref Expression
axrep2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axrep2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfe1 2014 . . . . 5 𝑤𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤)
2 nfv 1830 . . . . 5 𝑤𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))
31, 2nfim 1813 . . . 4 𝑤(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
43nfex 2140 . . 3 𝑤𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
5 elequ2 1991 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
65anbi1d 737 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑥𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
76exbidv 1837 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
87bibi2d 331 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
98albidv 1836 . . . . 5 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
109imbi2d 329 . . . 4 (𝑤 = 𝑦 → ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ (∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
1110exbidv 1837 . . 3 (𝑤 = 𝑦 → (∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
12 axrep1 4700 . . 3 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
134, 11, 12chvar 2250 . 2 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
14 sp 2041 . . . . . . 7 (∀𝑦𝜑𝜑)
1514imim1i 61 . . . . . 6 ((𝜑𝑧 = 𝑦) → (∀𝑦𝜑𝑧 = 𝑦))
1615alimi 1730 . . . . 5 (∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑𝑧 = 𝑦))
1716eximi 1752 . . . 4 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦))
18 nfv 1830 . . . . 5 𝑤𝑧(∀𝑦𝜑𝑧 = 𝑦)
19 nfa1 2015 . . . . . . 7 𝑦𝑦𝜑
20 nfv 1830 . . . . . . 7 𝑦 𝑧 = 𝑤
2119, 20nfim 1813 . . . . . 6 𝑦(∀𝑦𝜑𝑧 = 𝑤)
2221nfal 2139 . . . . 5 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑤)
23 equequ2 1940 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 = 𝑦𝑧 = 𝑤))
2423imbi2d 329 . . . . . 6 (𝑦 = 𝑤 → ((∀𝑦𝜑𝑧 = 𝑦) ↔ (∀𝑦𝜑𝑧 = 𝑤)))
2524albidv 1836 . . . . 5 (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑𝑧 = 𝑤)))
2618, 22, 25cbvex 2260 . . . 4 (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2717, 26sylib 207 . . 3 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2827imim1i 61 . 2 ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
2913, 28eximii 1754 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-rep 4699 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  axrep3  4702  axrepndlem1  9293
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