Theorem sb2 2425

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2038) or a non-freeness hypothesis (sb6f 2459). (Contributed by NM, 13-May-1993.) Revise df-sb 2017. (Revised by Wolf Lammen, 26-Jul-2023.)

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 1779	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥𝜑))
3		stdpc4 2020	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4	2, 3	syl6 35	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
5		sb4b 2424	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	biimprd 240	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑))
7	4, 6	pm2.61i 177	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1506  [wsb 2016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107  ax-13 2302

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-sb 2017

This theorem is referenced by:  sb3  2426  hbsb2  2443  hbsb2a  2445  hbsb2e  2447  equsb1  2452  equsb2  2453  dfsb2  2454  sbequiOLD  2456  sb6f  2459  sbi1OLD  2464  sbeqal1  40181