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Theorem sb6rf 2555
Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2380. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
Hypothesis
Ref Expression
sb5rf.1 𝑦𝜑
Assertion
Ref Expression
sb6rf (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3 𝑦𝜑
2 sbequ12r 2287 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
31, 2equsal 2437 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑)
43bicomi 216 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1654  wnf 1882  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220  ax-13 2389
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-nf 1883  df-sb 2068
This theorem is referenced by:  eu1OLD  2696
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