Theorem sbi1v 1812

Description: Forward direction of sbimv 1814. (Contributed by Jim Kingdon, 25-Dec-2017.)

Assertion

Ref	Expression
sbi1v	⊢ ([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 sbi1v

Step	Hyp	Ref	Expression
1		sb6 1807	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb6 1807	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
3		ax-2 6	. . . . 5 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
4	3	al2imi 1387	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
5		sb2 1690	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓)
6	4, 5	syl6 33	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓))
7	2, 6	sylbi 119	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓))
8	1, 7	syl5bi 150	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1282  [wsb 1685

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sbimv  1814