Theorem sblimv 1906

Description: Version of sblim 1973 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)

Hypothesis

Ref	Expression
sblimv.1	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sblimv

Step	Hyp	Ref	Expression
1		sbimv 1905	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sblimv.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
3	2	sbh 1787	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	3	imbi2i 226	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
5	1, 4	bitri 184	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by: (None)