Theorem sbanv 1913

Description: Version of sban 1983 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

Ref	Expression
sbanv	⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbanv

Step	Hyp	Ref	Expression
1		sb6 1910	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓)))
2		sb6 1910	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3		sb6 1910	. . . 4 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
4	2, 3	anbi12i 460	. . 3 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜓)))
5		19.26 1504	. . 3 ⊢ (∀𝑥((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜓)))
6		pm4.76 604	. . . 4 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ (𝑥 = 𝑦 → (𝜑 ∧ 𝜓)))
7	6	albii 1493	. . 3 ⊢ (∀𝑥((𝑥 = 𝑦 → 𝜑) ∧ (𝑥 = 𝑦 → 𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓)))
8	4, 5, 7	3bitr2i 208	. 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 ∧ 𝜓)))
9	1, 8	bitr4i 187	1 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  sban  1983