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Theorem spsbim 1724
 Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbim (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem spsbim
StepHypRef Expression
1 imim2 49 . . . 4 ((𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
21sps 1430 . . 3 (∀𝑥(𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
3 id 19 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
43anim2d 320 . . . . 5 ((𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
54alimi 1344 . . . 4 (∀𝑥(𝜑𝜓) → ∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
6 exim 1490 . . . 4 (∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)) → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓)))
75, 6syl 14 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜓)))
82, 7anim12d 318 . 2 (∀𝑥(𝜑𝜓) → (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓))))
9 df-sb 1646 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
10 df-sb 1646 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
118, 9, 103imtr4g 194 1 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 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:  spsbbi  1725  hbsb4t  1889  moim  1964
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