MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equs45f Structured version   Visualization version   GIF version

Theorem equs45f 2349
Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2289 and does not require the non-freeness hypothesis. Theorem sb56 2149 replaces the non-freeness hypothesis with a dv condition and equs5 2350 replaces it with a distinctor as antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
equs45f.1 𝑦𝜑
Assertion
Ref Expression
equs45f (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . . 6 𝑦𝜑
21nf5ri 2064 . . . . 5 (𝜑 → ∀𝑦𝜑)
32anim2i 593 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
43eximi 1761 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
5 equs5a 2347 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl 17 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
7 equs4 2289 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
86, 7impbii 199 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wex 1703  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1704  df-nf 1709
This theorem is referenced by:  sb5f  2385
  Copyright terms: Public domain W3C validator