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

Theorem sbtvOLD 2515
 Description: Obsolete version of sbt 2067 as of 6-Jul-2023. A substitution into a theorem yields a theorem. See sbt 2067 when 𝑥, 𝑦 need not be disjoint. (Contributed by BJ, 31-May-2019.) Reduce axioms. (Revised by Steven Nguyen, 25-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sbtvOLD.1 𝜑
Assertion
Ref Expression
sbtvOLD [𝑥 / 𝑦]𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbtvOLD
StepHypRef Expression
1 sbtvOLD.1 . . 3 𝜑
21a1i 11 . 2 (𝑦 = 𝑥𝜑)
3 ax6ev 1968 . . 3 𝑦 𝑦 = 𝑥
43, 1exan 1858 . 2 𝑦(𝑦 = 𝑥𝜑)
5 dfsb1 2506 . 2 ([𝑥 / 𝑦]𝜑 ↔ ((𝑦 = 𝑥𝜑) ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
62, 4, 5mpbir2an 709 1 [𝑥 / 𝑦]𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776  [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-10 2141  ax-12 2173  ax-13 2386 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator