Theorem stdpc4v 2244

Description: If a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Version of stdpc4 2428 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 31-May-2019.)

Assertion

Ref	Expression
stdpc4v	⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem stdpc4v

Step	Hyp	Ref	Expression
1		ala1 1857	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2v 2243	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1599  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012

This theorem is referenced by:  sbftv  2245  spsbimvOLD  2284  pm13.183  3549  bj-2stdpc4v  33346  bj-sbfvv  33352  bj-sbtv  33353  bj-vexwvt  33433  bj-ab0  33481