Theorem sbi2v 1772

Description: Reverse direction of sbimv 1773. (Contributed by Jim Kingdon, 18-Jan-2018.)

Assertion

Ref	Expression
sbi2v	⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbi2v

Step	Hyp	Ref	Expression
1		19.38 1566	. . 3 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) → ∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)))
2		pm3.3 248	. . . . 5 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦 → 𝜓))))
3		pm2.04 76	. . . . 5 ⊢ ((𝜑 → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → 𝜓)))
4	2, 3	syli 33	. . . 4 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → 𝜓)))
5	4	alimi 1344	. . 3 ⊢ (∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
6	1, 5	syl 14	. 2 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
7		sb5 1767	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8		sb6 1766	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
9	7, 8	imbi12i 228	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
10		sb6 1766	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
11	6, 9, 10	3imtr4i 190	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbimv  1773