Theorem sb5f 2489

Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2469 and does not require the nonfreeness hypothesis. Theorem sb5 2259 replaces the nonfreeness hypothesis with a disjoint variable condition on 𝑥, 𝑦 and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb6f.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb5f	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem sb5f

Step	Hyp	Ref	Expression
1		sb6f.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb6f 2488	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	equs45f 2450	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2, 3	bitr4i 278	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2363

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  sb7f  2516