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

Theorem sbievw2 2107
 Description: sbievw 2103 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 2106 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑)
2 sbievw2.1 . . . . 5 (𝑥 = 𝑤 → (𝜑𝜒))
32sbievw 2103 . . . 4 ([𝑤 / 𝑥]𝜑𝜒)
43sbbii 2081 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒)
5 sbv 2098 . . 3 ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
61, 4, 53bitr3i 304 . 2 ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑)
7 sbievw2.2 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
87sbievw 2103 . 2 ([𝑦 / 𝑤]𝜒𝜓)
96, 8bitr3i 280 1 ([𝑦 / 𝑥]𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  sbco2vv  2108  equsb3  2109  equsb3r  2110  elsb3  2122  elsb4  2130  eqsb3  2940  clelsb3  2941  sbss  4434
 Copyright terms: Public domain W3C validator