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Theorem sbievw2 2096
Description: sbievw 2091 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 2095 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑)
2 sbievw2.1 . . . . 5 (𝑥 = 𝑤 → (𝜑𝜒))
32sbievw 2091 . . . 4 ([𝑤 / 𝑥]𝜑𝜒)
43sbbii 2074 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒)
5 sbv 2086 . . 3 ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
61, 4, 53bitr3i 301 . 2 ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑)
7 sbievw2.2 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
87sbievw 2091 . 2 ([𝑦 / 𝑤]𝜒𝜓)
96, 8bitr3i 277 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  [wsb 2062
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 1908  ax-6 1965  ax-7 2005
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063
This theorem is referenced by:  sbco2vv  2097  equsb3  2101  equsb3r  2102  elsb1  2114  elsb2  2123  eqsb1  2865  clelsb1  2866  clelsb2  2867  sbss  4525
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