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Theorem ichnfimlem2 43126
 Description: Lemma for ichnfimlem3 43127: 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
StepHypRef Expression
1 nfs1v 2239 . . 3 𝑥[𝑏 / 𝑥]𝜑
2 nfa1 2123 . . . 4 𝑥𝑥 𝑥 = 𝑦
3 drsb1 2491 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑏 / 𝑥]𝜑 ↔ [𝑏 / 𝑦]𝜑))
42, 3nfbidf 2193 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑏 / 𝑥]𝜑 ↔ Ⅎ𝑥[𝑏 / 𝑦]𝜑))
51, 4mpbii 234 . 2 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
6 sbft 2235 . 2 (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
75, 6syl 17 1 (∀𝑥 𝑥 = 𝑦 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀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-12 2143  ax-13 2346 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770  df-sb 2045 This theorem is referenced by:  ichnfimlem3  43127
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