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Theorem bj-sbimedh 9911
 Description: A strengthening of sbiedh 1670 (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 1649 . . 3 ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦𝜓))
2 bj-sbimedh.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3 bj-sbimedh.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
43impd 242 . . . 4 (𝜑 → ((𝑥 = 𝑦𝜓) → 𝜒))
52, 4eximdh 1502 . . 3 (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑥𝜒))
61, 5syl5 28 . 2 (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒))
7 bj-sbimedh.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
82, 719.9hd 1552 . 2 (𝜑 → (∃𝑥𝜒𝜒))
96, 8syld 40 1 (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  bj-sbimeh  9912
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