Theorem sb5f 2386

Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 1883 and does not require the non-freeness hypothesis. Theorem sb5 2430 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

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 2385	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	equs45f 2350	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2, 3	bitr4i 267	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sb7f  2453