Theorem bnj90 34719

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj90.1	⊢ 𝑌 ∈ V

Assertion

Ref	Expression
bnj90	⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝑌(𝑥,𝑧)

Proof of Theorem bnj90

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj90.1	. 2 ⊢ 𝑌 ∈ V
2		fneq2 6613	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦))
3		fneq2 6613	. . 3 ⊢ (𝑦 = 𝑌 → (𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌))
4	2, 3	sbcie2g 3797	. 2 ⊢ (𝑌 ∈ V → ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌))
5	1, 4	ax-mp 5	1 ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  Vcvv 3450  [wsbc 3756   Fn wfn 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-sbc 3757  df-fn 6517

This theorem is referenced by:  bnj121  34867  bnj130  34871  bnj207  34878