Theorem sb2 2484

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2085) or a nonfreeness hypothesis (sb6f 2502). Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 13-May-1993.) Revise df-sb 2065. (Revised by Wolf Lammen, 26-Jul-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
sb2	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Proof of Theorem sb2

Step	Hyp	Ref	Expression
1		pm2.27 42	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → 𝜑))
2	1	al2imi 1815	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥𝜑))
3		stdpc4 2068	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4	2, 3	syl6 35	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
5		sb4b 2480	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	biimprd 248	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
7	4, 6	pm2.61i 182	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by:  hbsb2  2487  hbsb2a  2489  hbsb2e  2491  equsb1  2496  equsb2  2497  dfsb2  2498  sb6f  2502  sbeqal1  44417