Theorem spsbimvOLD 2328

Description: Obsolete version of spsbim 2077 as of 6-Jul-2023. Specialization of implication. Version of spsbim 2077 with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Wolf Lammen, 19-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
spsbimvOLD	⊢ (∀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 spsbimvOLD

Step	Hyp	Ref	Expression
1		nfa1 2155	. 2 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		sp 2182	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3	1, 2	sbimd 2245	1 ⊢ (∀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  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)