Theorem bj-ax12v 34837

Description: A weaker form of ax-12 2171 and ax12v 2172, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax12v	⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable groups:   𝑥,𝑡   𝜑,𝑡

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ax12v

Step	Hyp	Ref	Expression
1		ax12v 2172	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2	1	ax-gen 1798	1 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1798  ax-5 1913  ax-12 2171

This theorem is referenced by:  bj-ax12  34838