MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbievw2 Structured version   Visualization version   GIF version

Theorem sbievw2 2098
Description: sbievw 2094 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.)
Hypotheses
Ref Expression
sbievw2.1 (𝑥 = 𝑤 → (𝜑𝜒))
sbievw2.2 (𝑤 = 𝑦 → (𝜒𝜓))
Assertion
Ref Expression
sbievw2 ([𝑦 / 𝑥]𝜑𝜓)
Distinct variable groups:   𝑥,𝑤   𝑦,𝑤   𝜑,𝑤   𝜓,𝑤   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑤)

Proof of Theorem sbievw2
StepHypRef Expression
1 sbcom3vv 2097 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑)
2 sbievw2.1 . . . . 5 (𝑥 = 𝑤 → (𝜑𝜒))
32sbievw 2094 . . . 4 ([𝑤 / 𝑥]𝜑𝜒)
43sbbii 2078 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒)
5 sbv 2090 . . 3 ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
61, 4, 53bitr3i 300 . 2 ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑)
7 sbievw2.2 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
87sbievw 2094 . 2 ([𝑦 / 𝑤]𝜒𝜓)
96, 8bitr3i 276 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  [wsb 2066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067
This theorem is referenced by:  sbco2vv  2099  equsb3  2100  equsb3r  2101  elsb1  2113  elsb2  2122  eqsb1  2863  clelsb1  2864  clelsb2  2865  sbss  4467
  Copyright terms: Public domain W3C validator