Theorem sbh 1706

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 1696	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sbh.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.41h 1620	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
4	1, 3	sylib 120	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦 ∧ 𝜑))
5	4	simprd 112	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
6		stdpc4 1705	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
7	2, 6	syl 14	. 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑)
8	5, 7	impbii 124	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1287  ∃wex 1426  [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-sb 1693

This theorem is referenced by:  sbf  1707  sb6x  1709  nfs1f  1710  hbs1f  1711  sbid2h  1777  sblimv  1822  sbrim  1878  sbrbif  1884  elsb3  1900  elsb4  1901