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Theorem axprALT 5319
 Description: Alternate proof of axpr 5325. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
axprALT 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

Proof of Theorem axprALT
StepHypRef Expression
1 zfpair 5318 . . 3 {𝑥, 𝑦} ∈ V
21isseti 3514 . 2 𝑧 𝑧 = {𝑥, 𝑦}
3 dfcleq 2820 . . 3 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3503 . . . . . . 7 𝑤 ∈ V
54elpr 4587 . . . . . 6 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 339 . . . . 5 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
7 biimpr 221 . . . . 5 ((𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
86, 7sylbi 218 . . . 4 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
98alimi 1805 . . 3 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
103, 9sylbi 218 . 2 (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
112, 10eximii 1830 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∨ wo 843  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cpr 4566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-pw 4544  df-sn 4565  df-pr 4567 This theorem is referenced by: (None)
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