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Theorem sb5f 2517
 Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2493 and does not require the non-freeness hypothesis. Theorem sb5 2274 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb6f.1 𝑦𝜑
Assertion
Ref Expression
sb5f ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb5f
StepHypRef Expression
1 sb6f.1 . . 3 𝑦𝜑
21sb6f 2516 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31equs45f 2472 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
42, 3bitr4i 281 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by:  sb7f  2546
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