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Theorem sb6f 2389
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2356 and does not require the non-freeness hypothesis. Theorem sb6 2433 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sb6f.1 𝑦𝜑
Assertion
Ref Expression
sb6f ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6f
StepHypRef Expression
1 sb6f.1 . . . . 5 𝑦𝜑
21nf5ri 2068 . . . 4 (𝜑 → ∀𝑦𝜑)
32sbimi 1888 . . 3 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
4 sb4a 2361 . . 3 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
53, 4syl 17 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
6 sb2 2356 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
75, 6impbii 199 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wnf 1705  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by:  sb5f  2390
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