Theorem sb6f 2516

Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2494 and does not require the non-freeness hypothesis. Theorem sb6 2091 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb6f.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb6f	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sb6f

Step	Hyp	Ref	Expression
1		sb6f.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nf5ri 2194	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	sbimi 2080	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
4		sb4a 2499	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	3, 4	syl 17	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		sb2 2494	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
7	5, 6	impbii 212	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537  Ⅎwnf 1786  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071

This theorem is referenced by:  sb5f  2517  bj-sbievv  34553