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Theorem sb6rf 2487
Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2372. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker sb6rfv 2372 if possible. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)
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 2250 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
31, 2equsal 2435 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑)
43bicomi 226 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531  wnf 1780  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by: (None)
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