Theorem sb5f 1850

Description: Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.)

Hypothesis

Ref	Expression
equs45f.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

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

Proof of Theorem sb5f

Step	Hyp	Ref	Expression
1		equs45f.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	sb6f 1849	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1	equs45f 1848	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	2, 3	bitr4i 187	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393  ∃wex 1538  [wsb 1808

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-11 1552  ax-4 1556  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-sb 1809

This theorem is referenced by:  sbcof2  1856