Theorem ichnfimlem1 43670

Description: Lemma for ichnfimlem3 43672: 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	⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ¬ ∀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

Step	Hyp	Ref	Expression
1		naev 2065	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑏)
2		nfsb4t 2539	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (¬ ∀𝑥 𝑥 = 𝑏 → Ⅎ𝑥[𝑏 / 𝑦]𝜑))
3	1, 2	syl5 34	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑏 / 𝑦]𝜑))
4	3	imp 409	. 2 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
5		sbft 2270	. 2 ⊢ (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
6	4, 5	syl 17	1 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  ichnfimlem3  43672