Theorem bj-drnf2v 35992

Description: Version of drnf2 2442 with a disjoint variable condition, which does not require ax-10 2136, ax-11 2153, ax-12 2170, ax-13 2370. Instance of nfbidv 1924. Note that the version of axc15 2420 with a disjoint variable condition is actually ax12v2 2172 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-drnf2v.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-drnf2v	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem bj-drnf2v

Step	Hyp	Ref	Expression
1		bj-drnf2v.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	nfbidv 1924	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912

This theorem depends on definitions:  df-bi 206  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)