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Theorem wl-2spsbbi 37546
Description: spsbbi 2071 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.)
Assertion
Ref Expression
wl-2spsbbi (∀𝑎𝑏(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))

Proof of Theorem wl-2spsbbi
StepHypRef Expression
1 alcom 2157 . 2 (∀𝑎𝑏(𝜑𝜓) ↔ ∀𝑏𝑎(𝜑𝜓))
2 nfa1 2149 . . 3 𝑏𝑏𝑎(𝜑𝜓)
3 nfa1 2149 . . . . 5 𝑎𝑎(𝜑𝜓)
4 sp 2181 . . . . 5 (∀𝑎(𝜑𝜓) → (𝜑𝜓))
53, 4sbbid 2244 . . . 4 (∀𝑎(𝜑𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
65sps 2183 . . 3 (∀𝑏𝑎(𝜑𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓))
72, 6sbbid 2244 . 2 (∀𝑏𝑎(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
81, 7sylbi 217 1 (∀𝑎𝑏(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  [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  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by: (None)
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