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Theorem bj-ssbim 32959
 Description: Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2541. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbim (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Proof of Theorem bj-ssbim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 imim2 58 . . . . 5 ((𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
21al2imi 1891 . . . 4 (∀𝑥(𝜑𝜓) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
32imim2d 57 . . 3 (∀𝑥(𝜑𝜓) → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜓))))
43alimdv 1997 . 2 (∀𝑥(𝜑𝜓) → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜓))))
5 df-ssb 32958 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
6 df-ssb 32958 . 2 ([𝑡/𝑥]b𝜓 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜓)))
74, 5, 63imtr4g 285 1 (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629  [wssb 32957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991 This theorem depends on definitions:  df-bi 197  df-ssb 32958 This theorem is referenced by:  bj-ssbbi  32960  bj-ssbimi  32961
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