Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-sbimedh GIF version

Theorem bj-sbimedh 12780
Description: A strengthening of sbiedh 1743 (same proof). (Contributed by BJ, 16-Dec-2019.)
Hypotheses
Ref Expression
bj-sbimedh.1 (𝜑 → ∀𝑥𝜑)
bj-sbimedh.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
bj-sbimedh.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
bj-sbimedh (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Proof of Theorem bj-sbimedh
StepHypRef Expression
1 sb1 1722 . . 3 ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦𝜓))
2 bj-sbimedh.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3 bj-sbimedh.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
43impd 252 . . . 4 (𝜑 → ((𝑥 = 𝑦𝜓) → 𝜒))
52, 4eximdh 1573 . . 3 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑥𝜒))
61, 5syl5 32 . 2 (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒))
7 bj-sbimedh.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
82, 719.9hd 1623 . 2 (𝜑 → (∃𝑥𝜒𝜒))
96, 8syld 45 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312  wex 1451  [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-sb 1719
This theorem is referenced by:  bj-sbimeh  12781
  Copyright terms: Public domain W3C validator