Theorem ichnfimlem2 43671

Description: Lemma for ichnfimlem3 43672: When substituting successively for two always equal variables, the second substitution has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.)

Assertion

Ref	Expression
ichnfimlem2	⊢ (∀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 ichnfimlem2

Step	Hyp	Ref	Expression
1		nfs1v 2160	. . 3 ⊢ Ⅎ𝑥[𝑏 / 𝑥]𝜑
2		nfa1 2155	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
3		drsb1 2535	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑏 / 𝑥]𝜑 ↔ [𝑏 / 𝑦]𝜑))
4	2, 3	nfbidf 2226	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑏 / 𝑥]𝜑 ↔ Ⅎ𝑥[𝑏 / 𝑦]𝜑))
5	1, 4	mpbii 235	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
6		sbft 2270	. 2 ⊢ (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
7	5, 6	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([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-12 2177  ax-13 2390

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

This theorem is referenced by:  ichnfimlem3  43672