Theorem sb5ALT 40866

Description: Equivalence for substitution. Alternate proof of sb5 2276. This proof is sb5ALTVD 41254 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 = 𝑦 ∧ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5ALT

Step	Hyp	Ref	Expression
1		equsb1 2530	. . . 4 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		sban 2086	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
3	2	simplbi2com 505	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑)))
4	1, 3	mpi 20	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑))
5		spsbe 2088	. . 3 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
6	4, 5	syl 17	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
7		hbs1 2274	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
8		simpr 487	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)
9	8	a1i 11	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
10		simpl 485	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦)
11	10	a1i 11	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦))
12		sbequ1 2249	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
13	12	com12 32	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
14	9, 11, 13	syl6c 70	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
15	7, 14	exlimexi 40865	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)
16	6, 15	impbii 211	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)