Theorem ichnfimlem3 43672

Description: Lemma for ichnfim 43673: A substitution of a non-free variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.)

Assertion

Ref	Expression
ichnfimlem3	⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Distinct variable group:   𝑥,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ichnfimlem3

Step	Hyp	Ref	Expression
1		ichnfimlem2 43671	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
2	1	adantl 484	. 2 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
3		ichnfimlem1 43670	. 2 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
4	2, 3	pm2.61dan 811	1 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀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:  ichnfim  43673