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Theorem sb5f 2521
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2047 and does not require the non-freeness hypothesis. Theorem sb5 2565 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
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 2520 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31equs45f 2485 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
42, 3bitr4i 267 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1628  wex 1851  wnf 1855  [wsb 2044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-10 2166  ax-12 2194  ax-13 2389
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1852  df-nf 1857  df-sb 2045
This theorem is referenced by:  sb7f  2588
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