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Theorem ichnfimlem1 43125
 Description: Lemma for ichnfimlem3 43127: A substitution of a non-free variable has no effect. Give the distinctor in a form that can be easily eliminiated. (Contributed by Wolf Lammen, 6-Aug-2023.)
Assertion
Ref Expression
ichnfimlem1 ((∀𝑦𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
Distinct variable group:   𝑥,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ichnfimlem1
StepHypRef Expression
1 naev 2040 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑏)
2 nfsb4t 2495 . . . 4 (∀𝑦𝑥𝜑 → (¬ ∀𝑥 𝑥 = 𝑏 → Ⅎ𝑥[𝑏 / 𝑦]𝜑))
31, 2syl5 34 . . 3 (∀𝑦𝑥𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑏 / 𝑦]𝜑))
43imp 407 . 2 ((∀𝑦𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
5 sbft 2235 . 2 (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
64, 5syl 17 1 ((∀𝑦𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1523  Ⅎwnf 1769  [wsb 2044 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045 This theorem is referenced by:  ichnfimlem3  43127
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