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Theorem sb6f 2522
 Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2489 and does not require the non-freeness hypothesis. Theorem sb6 2566 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 2212 . . . 4 (𝜑 → ∀𝑦𝜑)
32sbimi 2052 . . 3 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
4 sb4a 2494 . . 3 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
53, 4syl 17 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
6 sb2 2489 . 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 1630  Ⅎwnf 1857  [wsb 2046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047 This theorem is referenced by:  sb5f  2523
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