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Theorem sn-el 39355
Description: A version of el 5247 with an inner existential quantifier on 𝑥, which avoids ax-7 2015 and ax-8 2116. (Contributed by SN, 18-Sep-2023.)
Assertion
Ref Expression
sn-el 𝑦𝑥 𝑥𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem sn-el
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-pow 5243 . 2 𝑦𝑥(∀𝑤(𝑤𝑥𝑤𝑧) → 𝑥𝑦)
2 ax6ev 1972 . . . 4 𝑥 𝑥 = 𝑧
3 ax9v1 2126 . . . . 5 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
43alrimiv 1928 . . . 4 (𝑥 = 𝑧 → ∀𝑤(𝑤𝑥𝑤𝑧))
52, 4eximii 1838 . . 3 𝑥𝑤(𝑤𝑥𝑤𝑧)
6 exim 1835 . . 3 (∀𝑥(∀𝑤(𝑤𝑥𝑤𝑧) → 𝑥𝑦) → (∃𝑥𝑤(𝑤𝑥𝑤𝑧) → ∃𝑥 𝑥𝑦))
75, 6mpi 20 . 2 (∀𝑥(∀𝑤(𝑤𝑥𝑤𝑧) → 𝑥𝑦) → ∃𝑥 𝑥𝑦)
81, 7eximii 1838 1 𝑦𝑥 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-9 2124  ax-pow 5243
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  sn-dtru  39356
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