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Theorem axrep4 4697
Description: A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
axrep4.1 𝑧𝜑
Assertion
Ref Expression
axrep4 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep4
StepHypRef Expression
1 axrep3 4696 . . 3 𝑥(∃𝑧𝑦(𝜑𝑦 = 𝑧) → ∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
2119.35i 1794 . 2 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
3 nfv 1829 . . . . 5 𝑧 𝑦𝑥
4 nfv 1829 . . . . . . 7 𝑧 𝑥𝑤
5 nfa1 2014 . . . . . . 7 𝑧𝑧𝜑
64, 5nfan 1815 . . . . . 6 𝑧(𝑥𝑤 ∧ ∀𝑧𝜑)
76nfex 2139 . . . . 5 𝑧𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)
83, 7nfbi 1820 . . . 4 𝑧(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
98nfal 2138 . . 3 𝑧𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
10 nfv 1829 . . . . 5 𝑥 𝑦𝑧
11 nfe1 2013 . . . . 5 𝑥𝑥(𝑥𝑤𝜑)
1210, 11nfbi 1820 . . . 4 𝑥(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
1312nfal 2138 . . 3 𝑥𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
14 elequ2 1990 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
15 axrep4.1 . . . . . . . . 9 𝑧𝜑
161519.3 2056 . . . . . . . 8 (∀𝑧𝜑𝜑)
1716anbi2i 725 . . . . . . 7 ((𝑥𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥𝑤𝜑))
1817exbii 1763 . . . . . 6 (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
1918a1i 11 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2014, 19bibi12d 333 . . . 4 (𝑥 = 𝑧 → ((𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
2120albidv 1835 . . 3 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
229, 13, 21cbvex 2259 . 2 (∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
232, 22sylib 206 1 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-rep 4693
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  axrep5  4698  funimaexg  5875
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