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Theorem sb6rfv 2365
 Description: Reversed substitution. Version of sb6rf 2480 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.)
Hypothesis
Ref Expression
sb6rfv.nf 𝑦𝜑
Assertion
Ref Expression
sb6rfv (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb6rfv
StepHypRef Expression
1 sb6rfv.nf . . 3 𝑦𝜑
2 sbequ12r 2251 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
31, 2equsalv 2265 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑)
43bicomi 227 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  eu1  2671
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