Theorem bj-ax12 32618

Description: A weaker form of ax-12 2046 and ax12v2 2048, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.)

Assertion

Ref	Expression
bj-ax12	⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12

Step	Hyp	Ref	Expression
1		ax12v2 2048	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2	1	ax-gen 1721	1 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704

This theorem is referenced by:  bj-ax12ssb  32619  bj-sb56  32623