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Theorem axrep4 4552
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 4551 . . 3 𝑥(∃𝑧𝑦(𝜑𝑦 = 𝑧) → ∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
2119.35i 1772 . 2 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
3 nfv 1792 . . . . 5 𝑧 𝑦𝑥
4 nfv 1792 . . . . . . 7 𝑧 𝑥𝑤
5 nfa1 2032 . . . . . . 7 𝑧𝑧𝜑
64, 5nfan 2064 . . . . . 6 𝑧(𝑥𝑤 ∧ ∀𝑧𝜑)
76nfex 2083 . . . . 5 𝑧𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)
83, 7nfbi 2070 . . . 4 𝑧(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
98nfal 2082 . . 3 𝑧𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
10 nfv 1792 . . . . 5 𝑥 𝑦𝑧
11 nfe1 1968 . . . . 5 𝑥𝑥(𝑥𝑤𝜑)
1210, 11nfbi 2070 . . . 4 𝑥(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
1312nfal 2082 . . 3 𝑥𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
14 elequ2 1951 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
15 axrep4.1 . . . . . . . . 9 𝑧𝜑
161519.3 2019 . . . . . . . 8 (∀𝑧𝜑𝜑)
1716anbi2i 717 . . . . . . 7 ((𝑥𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥𝑤𝜑))
1817exbii 1749 . . . . . 6 (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
1918a1i 11 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2014, 19bibi12d 330 . . . 4 (𝑥 = 𝑧 → ((𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
2120albidv 1798 . . 3 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
229, 13, 21cbvex 2162 . 2 (∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
232, 22sylib 203 1 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wa 378  wal 1466  wex 1692  wnf 1696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-rep 4548
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697
This theorem is referenced by:  axrep5  4553  funimaexg  5715
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