Theorem bj-drnf2v 37426

Description: Version of drnf2 2476 with a disjoint variable condition, which does not require ax-10 2176, ax-11 2192, ax-12 2213, ax-13 2404. Instance of nfbidv 1952. Note that the version of axc15 2454 with a disjoint variable condition is actually ax12v2 2215 (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 1952	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940

This theorem depends on definitions:  df-bi 210  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)