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Theorem sb6x 2459
Description: Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 2367. Usage of sb6 2081 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb6x.1 𝑥𝜑
Assertion
Ref Expression
sb6x ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6x
StepHypRef Expression
1 sb6x.1 . . 3 𝑥𝜑
21sbf 2258 . 2 ([𝑦 / 𝑥]𝜑𝜑)
3 biidd 262 . . 3 (𝑥 = 𝑦 → (𝜑𝜑))
41, 3equsal 2412 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
52, 4bitr4i 278 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wnf 1778  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-sb 2061
This theorem is referenced by: (None)
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