Theorem sb1 2499

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2272) or a non-freeness hypothesis (sb5f 2534). Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker sb1v 2091 when possible. (Contributed by NM, 13-May-1993.) Revise df-sb 2066. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
sb1	⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Proof of Theorem sb1

Step	Hyp	Ref	Expression
1		spsbe 2084	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2		pm3.2 472	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦 ∧ 𝜑)))
3	2	aleximi 1828	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4	1, 3	syl5 34	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5		sb3b 2497	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6	5	biimpd 231	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7	4, 6	pm2.61i 184	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1531  ∃wex 1776  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by:  sb3bOLD  2504  dfsb1  2506  spsbeOLDOLD  2507  sb4vOLDOLD  2509  sb4e  2520