Theorem sbt 2070

Description: A substitution into a theorem yields a theorem. See sbtALT 2073 for a shorter proof requiring more axioms. See chvar 2412 and chvarv 2413 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2069. (Revised by Steven Nguyen, 6-Jul-2023.)

Hypothesis

Ref	Expression
sbt.1	⊢ 𝜑

Assertion

Ref	Expression
sbt	⊢ [𝑡 / 𝑥]𝜑

Proof of Theorem sbt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbt.1	. . . . . 6 ⊢ 𝜑
2	1	a1i 11	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝜑)
3	2	ax-gen 1795	. . . 4 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)
4	3	a1i 11	. . 3 ⊢ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	ax-gen 1795	. 2 ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		df-sb 2069	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	5, 6	mpbir 233	1 ⊢ [𝑡 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1534  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795

This theorem depends on definitions:  df-bi 209  df-sb 2069

This theorem is referenced by:  sbtru  2071  sbimi  2078  vexw  2804  vjust  3494  iscatd2  16948  iuninc  30312  suppss2f  30384  esumpfinvalf  31359  wl-rgenw  34881  sbtT  40975  2sb5ndVD  41318  2sb5ndALT  41340