ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  alexeq GIF version

Theorem alexeq 2856
Description: Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. (Contributed by NM, 2-Mar-1995.)
Hypothesis
Ref Expression
alexeq.1 𝐴 ∈ V
Assertion
Ref Expression
alexeq (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem alexeq
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 alexeq.1 . . 3 𝐴 ∈ V
2 eqeq2 2180 . . . . 5 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
32anbi1d 462 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
43exbidv 1818 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
52imbi1d 230 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
65albidv 1817 . . 3 (𝑦 = 𝐴 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
7 sb56 1878 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
81, 4, 6, 7vtoclb 2787 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑))
98bicomi 131 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by:  ceqex  2857
  Copyright terms: Public domain W3C validator