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Theorem bj-ssbssblem 33155
Description: Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbssblem ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)
Distinct variable groups:   𝑦,𝑡   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssbssblem
StepHypRef Expression
1 bj-ssb1 33140 . 2 ([𝑡/𝑦]b𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 bj-ssb1 33140 . . 3 ([𝑦/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
32bj-ssbbii 33131 . 2 ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b𝑥(𝑥 = 𝑦𝜑))
4 df-ssb 33127 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 43bitr4i 295 1 ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  [wssb 33126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-11 2200
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-ssb 33127
This theorem is referenced by: (None)
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