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Theorem sb5ALT 39048
Description: Equivalence for substitution. Alternate proof of sb5 2458. This proof is sb5ALTVD 39463 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5ALT ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5ALT
StepHypRef Expression
1 equsb1 2396 . . . 4 [𝑦 / 𝑥]𝑥 = 𝑦
2 sban 2427 . . . . 5 ([𝑦 / 𝑥](𝑥 = 𝑦𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
32simplbi2com 656 . . . 4 ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑)))
41, 3mpi 20 . . 3 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
5 spsbe 1941 . . 3 ([𝑦 / 𝑥](𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
64, 5syl 17 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
7 hbs1 2464 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
8 simpr 476 . . . . 5 ((𝑥 = 𝑦𝜑) → 𝜑)
98a1i 11 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) → 𝜑))
10 simpl 472 . . . . 5 ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦)
1110a1i 11 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦))
12 sbequ1 2148 . . . . 5 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1312com12 32 . . . 4 (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
149, 11, 13syl6c 70 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
157, 14exlimexi 39047 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
166, 15impbii 199 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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