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Theorem ichbidv 43896
Description: Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.)
Hypothesis
Ref Expression
ichbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ichbidv (𝜑 → ([𝑥𝑦]𝜓 ↔ [𝑥𝑦]𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem ichbidv
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ichbidv.1 . . . . . . . 8 (𝜑 → (𝜓𝜒))
21sbbidv 2085 . . . . . . 7 (𝜑 → ([𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦]𝜒))
32sbbidv 2085 . . . . . 6 (𝜑 → ([𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑦 / 𝑥][𝑎 / 𝑦]𝜒))
43sbbidv 2085 . . . . 5 (𝜑 → ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒))
54, 1bibi12d 349 . . . 4 (𝜑 → (([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓𝜓) ↔ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒𝜒)))
65albidv 1922 . . 3 (𝜑 → (∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓𝜓) ↔ ∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒𝜒)))
76albidv 1922 . 2 (𝜑 → (∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓𝜓) ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒𝜒)))
8 df-ich 43889 . 2 ([𝑥𝑦]𝜓 ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓𝜓))
9 df-ich 43889 . 2 ([𝑥𝑦]𝜒 ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒𝜒))
107, 8, 93bitr4g 317 1 (𝜑 → ([𝑥𝑦]𝜓 ↔ [𝑥𝑦]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  [wsb 2070  [wich 43888
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 1912
This theorem depends on definitions:  df-bi 210  df-sb 2071  df-ich 43889
This theorem is referenced by:  ichim  43900
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