Theorem sbequ8 1847

Description: Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)

Assertion

Ref	Expression
sbequ8	⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑))

Proof of Theorem sbequ8

Step	Hyp	Ref	Expression
1		pm5.4 249	. . 3 ⊢ ((𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑥 = 𝑦 → 𝜑))
2		simpl 109	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) → 𝑥 = 𝑦)
3		pm3.35 347	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) → 𝜑)
4	2, 3	jca 306	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) → (𝑥 = 𝑦 ∧ 𝜑))
5		simpl 109	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦)
6		pm3.4 333	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
7	5, 6	jca 306	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)))
8	4, 7	impbii 126	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ↔ (𝑥 = 𝑦 ∧ 𝜑))
9	8	exbii 1605	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑)) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	1, 9	anbi12i 460	. 2 ⊢ (((𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑))) ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
11		df-sb 1763	. 2 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑) ↔ ((𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → 𝜑))))
12		df-sb 1763	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
13	10, 11, 12	3bitr4ri 213	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1492  [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  sbidm  1851