Theorem sbequ1 2244

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2066. (Revised by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
sbequ1	⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))

Proof of Theorem sbequ1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equeucl 2027	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑦 = 𝑡 → 𝑥 = 𝑦))
2		ax12v 2173	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	1, 2	syl6 35	. . . 4 ⊢ (𝑥 = 𝑡 → (𝑦 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
4	3	com23 86	. . 3 ⊢ (𝑥 = 𝑡 → (𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
5	4	alrimdv 1926	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
6		df-sb 2066	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	5, 6	syl6ibr 254	1 ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1531  [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-12 2172

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

This theorem is referenced by:  sbequ12  2248  dfsb1  2506  dfsb2  2528  sbequiOLD  2530  sbi1OLD  2538  2eu6  2740  bj-ssbid1  33992  sb5ALT  40852  2pm13.193  40879  2pm13.193VD  41230  sb5ALTVD  41240