Theorem sbi1vOLD 2323

Description: Obsolete version of sbi1 2076 as of 24-Jul-2023. Move implication out of substitution. Version of sbi1 2076 with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbi1vOLD	⊢ ([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 sbi1vOLD

Step	Hyp	Ref	Expression
1		sb4vOLD 2096	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
2		sb4vOLD 2096	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3		ax-2 7	. . . 4 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
4	3	al2imi 1816	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
5		sb2vOLD 2097	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓)
6	2, 4, 5	syl56 36	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7	1, 6	syl 17	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by:  sbimvOLD  2325