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Theorem axext3ALT 2753
 Description: Alternate proof of axext3 2752, shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axext3ALT (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axext3ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2064 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 336 . . . 4 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1879 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 equequ1 1982 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 337 . 2 (𝑤 = 𝑥 → ((∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
6 ax-ext 2750 . 2 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
75, 6chvarv 2327 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-12 2106  ax-13 2301  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747 This theorem is referenced by: (None)
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