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Theorem spsbim 1740
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 53 . . . 4 ((𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
21sps 1446 . . 3 (∀𝑥(𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
3 id 19 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
43anim2d 324 . . . . 5 ((𝜑𝜓) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
54alimi 1360 . . . 4 (∀𝑥(𝜑𝜓) → ∀𝑥((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
6 exim 1506 . . . 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 322 . 2 (∀𝑥(𝜑𝜓) → (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓))))
9 df-sb 1662 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
10 df-sb 1662 . 2 ([𝑦 / 𝑥]𝜓 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
118, 9, 103imtr4g 198 1 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397  [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:  spsbbi  1741  hbsb4t  1905  moim  1980
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