Theorem spsbimvOLD 2251

Description: Obsolete version of spsbim 2023 as of 6-Jul-2023. Specialization of implication. Version of spsbim 2023 with a disjoint variable condition, not requiring ax-13 2301. (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 2088	. 2 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		sp 2111	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3	1, 2	sbimd 2172	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1505  [wsb 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016

This theorem is referenced by: (None)