Theorem sbh 1749

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)

Hypothesis

Ref	Expression
sbh.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
sbh	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Proof of Theorem sbh

Step	Hyp	Ref	Expression
1		sb1 1739	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sbh.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.41h 1663	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
4	1, 3	sylib 121	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
5	4	simprd 113	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
6		stdpc4 1748	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
7	2, 6	syl 14	. 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑)
8	5, 7	impbii 125	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sbf  1750  sb6x  1752  nfs1f  1753  hbs1f  1754  sbid2h  1821  sblimv  1866  sbrim  1927  sbrbif  1933  elsb3  1949  elsb4  1950