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Theorem spsbbi 1741
Description: Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbbi (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem spsbbi
StepHypRef Expression
1 spsbim 1740 . . 3 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 spsbim 1740 . . 3 (∀𝑥(𝜓𝜑) → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑))
31, 2anim12i 325 . 2 ((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)) → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
4 albiim 1392 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
5 dfbi2 374 . 2 (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ∧ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜑)))
63, 4, 53imtr4i 194 1 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbbid  1742  relelfvdm  5232
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